83
Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside or on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 Citation for published version (APA): Bouwkamp, C. J. (1966). Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside or on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000. Philips Research Laboratories. Document status and date: Published: 01/01/1966 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 12. Dec. 2020

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Page 1: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

Number of lattice points on the surface of a sphere of radiusSqrt[N] (N=1 (1) 10000 . Number of lattice points inside or onthe surface of a sphere of radius Sqrt[N] (N=1 (1) 10000Citation for published version (APA):Bouwkamp, C. J. (1966). Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 .Number of lattice points inside or on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000. Philips ResearchLaboratories.

Document status and date:Published: 01/01/1966

Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne

Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.

Download date: 12. Dec. 2020

Page 2: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

------~--------------------

____ tlUJ~LÛi~ ~ ~,~~s--~-~-~-1 Q_~-o::f~-1/;J_~

-------:----· d-\~~--~---~- 4~ . ~ vv G"h- -~~~~-~N -

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------ --L~ - t _ ______:__N~U_:__~ -=--l ~ ~ N :::: i(~~-o-~_Q_9 ---~Lto-~}-,....

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~~--> ~~ _L~~ G_' ---

Page 3: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

NJJML ..

--~····--········----··-·-··--1:~: ·i;- - ---i! {ti ~:: t:i~ ,~-- -:fi~- 1;;~ .

8 '3 ... ---· --·· _____ 1..2... . ....... ill~- .. _ _ ____ ft._ ---·---··-.J..J.3__ __________ -U.4-:-__ -- ------2-03- ... U:. -6· 54 96 ig4 '12 1!14 06 204 48

.. 2.4.~. __ e_ _ . _____ o_··-----·----·-----~--- _JA.____ __UJ- ... u. 2os ,,. 14· !56 48 tÖ6 '12 1!16 0 206 2~0 Jt !7_ 4.8. tö..i · -- '11 t.•'1 -,.. I ó-'1 ··~

' 8 12' 58 24 158 32 118 06 !Ó8 14 _ _ __ ;_ . __ 3.a_ -·-- ___ lt .?a___ --IÜ-- --12- --·1-19- · --t- 2-0-9-- · - ··· e•oc

iö ••· 6o o tio !44 16o •• etä t6 -------------------------------------- -- -11· - .... a-i: .. ---- :t " ---it ·- --------·jii: · - --:-- · i!i- -i::: ii~- ;:-·-···-·-·-·-··-··---·-·-------------· _____________ u_ .. _____________ _a_4:_ ________ ll.:_ __________ a. ________ ---UJ- _______ -- _..__ __ - -- ----t-•3--- ------'-l-4-"-- --- --2-1.3----·-··-- - . -·-U-- ..

14 49 64 6 ti4 98 164 06 214 ., •. -- - --- -- ---- ------- - ------ - __ ..11.:__ --- ___ _a_--- - __ u__ --- - -- 9_6 -- --- - ---i-!!---------------- _u_ ---~U5-- -- - -- ------4---- - - -214--- - - ---G-- -

16 . 6 66 96 116 72 166 110 116 ., _ ··-···- ____ "_.1 __________ " __ !_!:..____ -------~~? ________________ 1.4 ___________ ----~! '~----- ... .s.t.a ____________ uL---·-··----·- --·--------~!_____ 1-t'l--· -u ..

18 36 68 48 ,,. 72 168 48 118 120 -·------------- -- - _ _1l_ __________ ~j __________ _j~------·-·-·-·--·-·----'__6_ --- ----·--•t.t: ____ , ... --D----····--·----16_Q ____________ "_,_. _______ .,,,,, -2.~-- ~·-

'' 25 24 70 48 titi 48 170 144- 115 ó 21. 4!___ .7.1.:---------0--- -ii-1· '-•---- 1'1 -t•o--- -ltt- - to-t 22 24· '12 36 112 SI& 112 24 222 144 23 ... ~--- .... ~~-.D.: .7.3 48. -H3--~ U- - .. t-»-- . 16&: . ft3:- .. . 0 24 24 74 120 !14 0 174 144 214 41 2.!__ ___ u.· !s_ t6 _ !P. s«- _....,_, -ö- -- 2-ts-- :t.fil -- -26 12: '16 24 116 i44 176 14· !26 96

~ ----- " ................. 1-~-

6

. 2.' - .32.: "11.' 96 i-P-~ ----.e- - . --!.-1-'1- -4-&· - "'- - 1-fl--·· 21 ö 78 48 lis 12 178 t6 218 48 29-- . -------7-2-: . -'10 .. ---{)-- - --------i-tt·--- ·---- t44 - 1,9 ----~- .. -t-t-0--------- - -~»-----------· 1fl -. 3~ 49 eo 24· 1~0 48 11ó '' 2sö 24o 31 0 81 S.-02' -tS!' -·- t-2-t- 111 11-G - - - !St~ ·-t 32 12' 82 48 i32 48 182 144 2tl2 14

_ --- ... 33 -4Ac. ···· 8-J f-!1··-- 41 ·- ·1-13 · ·- -Q- IS!'-··· t44·

_______ ll, ______________ 1t _ -----~-__:: ______ -~--- .. ::.... . .. Ut--- .!~- - i:; ---1::: :.:; 2~~, 36 30 86 120 t~6 48 116 144 236 ,.

--------- - -- --· ----------- -. -· ----------~31' ....... --- .!.4- - - ______8J _____________ __()_ ___________ ..._,..:_ ______ -- --'-6- -- ----··1-1-'1- --41~- .,_,.__ .. ! ••• ,......-38 12: 88 24 t~8 96 188 0 238 06 39 __ a_ ____ U___ -- ... 1.44 --- . -119·-- - --71·- ..... ----~.t.f tOl 219- Ö 4ö 24 90 120 t4ö 48 100 48 24Ö 0 'li ___________ -----~--~- ....... 1.1........ . .. 4 .. 8... .. i~2.~ 96. 111 0 241 144-42 49 92 0 ~~· 48 112 8 2162 131 43 ~!·____ 9:5_ .... !1\S 1A3 0-- - 1.U U 2163:. 104· 44. 24· 94 96 !44 . 38 104 240 244 '2 4!l .,.2: 9!. 0 tÁ-!l 96-- . 1 ._, . .... 24§ 168· 46 45 96 24 1.6 !92' 106 14 246 144 .~'l... . __ ___a_ .. 9..1_ 1UL. _ .i-47. .!16. - t-•'1- -llt- U'J 0 48 e 98 s.oe i4a 24 1t1 110 a•s ''

.. At. _ . '4: _9.9 12 ~~,, ~·•- s.-o; . -.o a•t 144-5o e•- îoo 30 ï•o ete aöo ••· 21o 144-

Page 4: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

.. Ji ... NUt11

---~-~_1__ 168 101· 96 ~!l _0__ . -~11 ......... J.!.fL~----- 41..1 ___ ........ 1.41_ 2!52 ··r··-------,o-2~- 144 312 24 4öa· 1•2, 4Ba· ' ' 2!53 4!)· i03 0 :!!Î:S __ $~2 403 .... !8 _____ . il3; ········· 144

--------~---·-···-------·----------~--~----254 ictr··--·--- •o4··- --·-·a4 384- s•a ·;.-a4 1•• 414 161

··-- •····· --~i;:- -i, --1~~ i-~~-~ ~-1! ,:: ..... :~; ~i:· !·:: 9~ ------------ -- ---------------- -- -~i~1- ·· - t:r- }~~ - ~l ~SI --~: :1: 4: -::~ ~::.

2,_, 9~- .... 10.9........ 1~.4 !it .. ·----. ·- .. - 4ÎÎ9 -- 102: 419. 101' 260 96· S10 96 3~0 120 410 102 460 48

----------------------------------·--· ----------------------2-6.1 2.1~..i.. __ ... - __ J.U_: ____ - -- Q 3.&.1.. .UIL - - 411- 1-'-4· . 4-61: 360 262 12. t12 48 :!62' 216 412 0 462 ,,.

----------------------------·---·-----··-----------------·--····2!t3 .... _________________ 11_ __________ ~--------.-t6 ....... ------36! ... - -- . -----10-4 '--- .. . ..... -4-1--!--- .. ~:....... .. .. ----46J-;...... -·-· 0 .. 264 ,, 114 312 3.4 48 4{4 24ó 464 ,.:

--·····----------·---------------------··-:.U!l---------''-'-- · !1!! ---1«------- 361'------- 240 ·---4-tc.!l----- ··· --ê----~---... U-'----------1-ff-L ····· 266 240: .16 0 366 l44 4i6 ,, 466 96

...... :.~--------------------AL--------· -U-1-----------· --11-0------· -- 3~--- · -- -e.- --4-1' !44- - ·46!1 16--8 268 24 818 144 !68 D 4{8 ·96 468 1!0

.. !6Q _______ !.6.4· _________ J12. .. _ --- - _(). ________ ... ·-· w .. ----·-- 2--81----· - ---4t9 216--c - ···------460 191' 2?Ö 19! 820 24 310 !44 410 06· 416 240

.. 2'li _________ 0__:_ ______ !2L ··------------ .2.4.0 .......... -!?1 .t92-~--- 4!1 . - 110 4-1i- ·0 ---···-· a7a •• 122 96 3?2 4& 422 120 ''2 12 a.7.s •s.. .12:s ~ .... i.6 _ 373: a2e 413 _ o.. 4!S- 1!44-2?4 144 aa• 102 sf4 a36 4!4 12 414 240

-~2!!L_ _ ___ 12_0 __ _ .. 5.2.! _ .ua ~!-?1-- a __ 42.!~ 3.36. .,,. 12~ 276 ,,, t!6 264 !16 ,, 416 218 416 0

... 2.11 _ 12: _ 1-21. .n _ sl1 __ tt.t .4U •a 4'17 21& 278 168 828 48 3f8- t92 418 ,. 418 96 2'!9 Q_ -129 288 :J,-9 . . '1 4-tt- .. 1'-2 - _,,, 0' 28o 48- s:so '' 310 e 430 144· 41Ö 48·

...... a.et ___ a•o _ ~-U1 12 sa1· a•o~- ... •:u. . _ 0- -- -••i 1oa· 282 06 t:sa 12 312' 06 432 sa· 482 24Ö

-~--------- ~-------- __ .. 2.~t:L .. ________ _tr_2_~----------S3.3.:.. .. --- -- --120-- ------ ---!U---- . - -----0- -- . 433----- - .......... t-4-4- -·-·- 4-ts· 96+-· 284 0 j:S'4 144 384 24 4!4 288 484 ,.

----------- __ 2.8JL ____ ~-------- .19.,L _____ ----- 3.!.:. _____ .... -· --0------ - --.SàJ-- -U - -4-D, 06- 48' 24è· 286 144· s:s• 4e ,,, a4e 4!6 '• 486 s11

·------ _QL_ ______ ,:J"J__ ______________ _., ____ -----U-1-'-----------1-1-t----------43--1-- -------····4•t-~-------·~--- ·······--·····-······0·+-288 s6i S38 110 !81 •a 438 o, 48e 120

-- - -- --- -- ---- --299 ~- .... ----1-!1.2.~-- . pg_ ---------- --1-4-4- . -!8-9----- -- -- 264- ···---· . --439-·· .. ··-······-·---0- -------- -48-9----- --14 & 29Ö 240 S40 48 !00 !92 440 144 49Ö 168 29-l - - •~-'- t•s. .s:.ur- stt.· _& 444. 210 •of at& 292 48 142 216 !92: 208 442 16 492 48 .293. . -2!S t•:s -0- -3'0! -l44 44-3- - 120 49! 144-294 165 ~44 120 !14 SlO 444 0 494 336 t9!f _ o _ . ~~!l 96.. s•s' at a . 44S ••~ 4t!f ó -296 120 146 120 396 72 446 314 416 0

... '2!!'L_ 19~ ~~7. 120 · u1· 11 447 o 4t' aaa 298 72 148 (I !08 140 448 0 408 96r .299 .... 1~_a· !it9 168 sot .... - O- - . 440 240 4to 71: sö5 !6 j5o 240 4ÖO so 410 212 so6 1~4

Page 5: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

. N- . NU-M

-~~-~-~~-~-----~!li.i_ .. -----~_ij2~ 15 -6Ü . 248---· .6.!!11 ~-· -1'-2· -·····- -'0!:~ --~-408~ ----~~-----~---···~--~-~-·--·-~~~~~---·---·~-~---~--;ll 1't 1:t . :! :lr ::: .!i~~ Jt~~ 11:- 1':

so4 1~4 154 264 664 c 614 336 704 a4 -~lÖ! . 9..&· U!· . .tA... .4-iJ: 112 - ·-'" o 7QB· aa-t !106 336· 156 72 606 S44 616 16, 706 2!8

-~--··-···~·-······-·· .. .!Q1 .. 1fU' .!51... . ~.2.16. ... .. 6.Ö-7' -0--- -- 61'1 ci4-Q.--- --·· - ·· 1 Ó' ·&«· !ÖI 0 ~18 288 6Ö8 1! 618 06 108 48-

_!..Q_I _ ..3 6.0 ~ -- -~ .UL -~-64.t U-1- .61-9 tt.4· - 70! t U· -Sf6 192: 160 48 6!0 !44 660 06 1!0 384·

·· · !51i .Ó..:. ·· f'-1. · ·· · ila.. 6Ït' a4(Jc. ·'-'' 216e J~1· ·f} 512 !2: 162 96 6i2' S44 662 264· 1t2 96

-----~-- ··-------··-- -~--~-----------~---.--5-U. -~------~L----~--~..IU~-----~2.1.6... 4-Ï.S---··· ·····---14 ··--····-- ·U3--- - -·--···---~-c--· -- - ---1-!.'5--~ ·· ·-lü---514 192 964 16 6!4 408 664 120 114 211

--···-·-··---- ---~----- -------~----~-------liL-------tij..__ _____ --!.6.! ___ . ------1-4-'-'---·-····---6Î5 1 -----------&· ·------~6J-·----- - ·tl-I.:. ---- ··-11-'--- ---·- ·····f-6· 5i6 1~4- 166 360 6i6 ,, 666 360 ,16 t2Ö a:~-.7. .. --1•'~-- ..... Ul ... ---0-- ....... --li-J------- - . ~a••· --u?- ··· ---46-c·· · 1t'· -t-t-t· 518 19•: 168 .48 ,,a s•• 668 o 'ia 144

·-----·---.. ··-~J.iO .. __ ..... ~-·------~.n.'--------~-!U-----~----U-4. -·- -- ----6-tO----- ... --- -th- ··· - - · fl-49 ·······-··· --- ~·'*'· - ···-- 1f9 0 · ·525 49 t'1o 112 &tri t6 61l'l 144 12ö 1t·

. S2i ---- .!lt4·- - -~!J7-1--- -1!0~- 6it. -- 184·- -- ~--·•'t' - &,- 111· 1•1 !522 120 172 0 6à2 !44 612 48 112 218 523 . uo . 113 192 ".,, f .• ,3 !44· ,.3 .,. 524 120 '14 112 6Q4 G 614 288 '?14 120 52!!J 24.0. .17.5 0 6t! . S!.O- 61!5 214" ,1!5 360 526- 144· ,,, 30 ••• 432 6~6 '18 ,,, 264

.. !12'1 a _,,, - 96 u1~ . __ ,,, . -6'1'1 -360 ,,.,. it 528 49 178 204 628 12 6'?8 240 '?18 144 520 1!!5-0 ,_,,. 102- ·~ -ü!-- . .,,,. --e· ,.... 3!1 !130 336 ,.0 96 630 148 680 144- 13Ö 144 531. -- 21.6. l81 33.6. 631 8 · 6&1 c240 711 ~~~ . 532 4 8 182 192 6~2 06 682 144 132' ö

___________ JJ3J __ ------~·.4·~ --·--- _J8.3 ·----0 ··-· -633- -2-48- - 6f3--- ·11! 133;- t6t: . 534 240 184 192 6~4 !68 684 1tó 73• 48ó

.... !3.!. .. ----······-- ----.IL _____ ---· .. i.U ...... - ....... z_IIL.. ~;iJ -340 -6-IJ;: ··1.•4· .· 11& 0 536 1~9 JB6 216 636 0 686 384 136 48

.!537 ____ -~-1.U:.~----~-.!~8.2. .. ________ 1JA _______ ..... ,,: .. $61------- ·····~--6·1 ···········~··· . '111~- ··240· --- -538 12Ó ,.8 16 631 348 688 14 ,38 240

...................... _.539___ ... zi_i!,_ _____ ~---!19 ... _____ 112 -- --ut·--· --- -..á-----~ 619---- ..... 4-10- 139-' tto 546 ó ,,0 240 6~è 24 600 191 '?45 ,., 54L 12J.L . 19-1-- o .. 6~1 136 691 110 741· 1:88 542 28!· 992 24 6~2 192 692 168 '?42' ,,, !143 - 0. J93 ..... 2!8. 643- 12 -t.03 . -211 '?43: f) 544 4! 194 314 644 !92 614 110 744 1~4 .

_Jts lJ4 . •t! 96 ,~,. ••a . ''' ~- '?45 1t1 546 2ss tt6 168 6•• sta 6t6 144 746 312

. .54' __ '12~ J91 144 •-'' e ••7 "'"" '1~!1 216 548 IS t98 96 648 !08 618 Stl: 148 48

. 5.4..9_ _ 3.61L 1.99 ... 0 . 649. 244 6.0.9. 240 ,49 · Sl4· 555 16! l~O 120 6.0 16G 1ÖO 0 '16 288

---· -· __ " _______ ,,.~ ... -~--- .............. .

Page 6: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

------~--------~----- ------~--~~~--~--~----·---~-~-- ---:: ::::-~-------- 0 ___ J_D 1___ __ 4 ~-2 _ __ _ 8.Î1.~ 2~JL ~U.·--~_ _ _ __ 21.1.:___ _ ___ Ut.:_ ~ ___ 4 752 0 102 144 IÎI· 96 902 336· 911 96

----~--------------------~--- ---------''5:1 L4~'-- ____ j_QJ~ ___ 2.!0. _u_~_ liG ~.Q.fi.J --tL.. ..9.13:_ Ja~--754 a-o 1o4 144 ei4 118 oa• t6 ••• s&o

____ . _____ J..!J. Ut· ... to.5. 192: at.S-:. a ._;g!l; a••· . ou.· ••· 756 192· !06 336 816· 72 906 :536 9!!J6 D ''" 120 so1 _ o u1· 11' _,~,- 1.1 ui· 1-12 ---------,,--8- '2&4-:- . 808 -- '?2 8!18 191 908 11Ó- 918 191 759 0 809 384 IÎ2 _i61 9.Q9. . !b.4 tlo· 0 160 •fi il(f 408 860 o oio 101· 96à o

-----------~------------------------------· --------- -l~~ 1!-~--- -- --~i i- 1 ::- :!i· a:}- ----:i; --- -,-f-- - ·:!i· i:!: _____ -------~----------------------------·- ______________ ·---~-~---------------1:-~_:5 ______________________ 1!_!__ ___________ 113 ___________ l~~ --~- _ _ _ __ S_._I,__ ___ _ ______ .o_________ _913~ ~ ____ . _ J.-''-·--- _________ 1.63. ... - ~ ---~ . ~11~'---~

164 o· 614 144 864 96 9!4 432· 964 144 -----~----~-----------------------------------------ZA~---------a•o ________ ll~--------------.ll- ... ------W·--- _______ t.n ________ tl.s~ . ___________ . .tt•: ____ --~----'•''--- -~~~ ...... Jta.

766 281 $16 48 866· 128 916· 110- 966 218 _________ !_6..'"------- uz ____________ i.t4 _ U-'-·- .tla.. ti'1 tllö. · _,,, ---~ó--

' 68 s u. a 336 868 96 oie 1012: 968 t3t· __________ t6..2 _____________ l__._o.__ ____________ l1..9 _________ _a!!.o .............. Ut.~- 1.1'----~~~-~~~ -~ tit· --~ .. -- .... cO· ~---1-60~----- 218·--~

775 J84 120 96 87ó st2 920 a•o 976 1~• 17i .1~4~ ...l2.1 ·--- - ~ !60 Û1: - --- -.&. 911 --240 .. - 0.'1-1• :SlOc 112 49- 122 24o 8;a· seo 112 216· 112 1o•

.. 113 .na~ - -- -tl!- . ----0-- a;J: a•• . ns:-- . . a•ö- - ,,,.- t-4*· 174 360 124 240 8,4 248 '"4 ó 914 431·

. I1!l ---- -- .0--- ~---U! 3-36 - ~an1 UI-- JU -- -141 9'1, -d-776 240 126 144 .,, 96 916 41ó ''~ , •.

. 1..7'! 19.2.:. ..821 168 ,,,. t-10-- -- - "'-- 0- ,,, 2140 ''8 161 128 o et• 24e 9ta •• ''8 12aa 1.79 . -2~0- ---tU----- -- 2..6-4 1'1t -----1 . -- ~- Ut- - 43l'~- -0'19 l-91: '186 96 è30 240 880 IJ 930 218- 91Ö 168 'lBi. __ a~a. --.!31- . o _ 8-at _ _,., -9-31'~ 1-'1 tlt :S!O 182 288 132 24 882 124 932 144· 912' 110

--~-----------------······------~-~- _1~3- ----- _____ n_ _________________ _ll_! _______ 3_3_6__ .• 1_!: ______ ~------ 1-2 ....... ------9.!3:...... ~lil:·- ----·.oa:s:::--- . ··-0-'?84 !J4· 134 192 884 !192 934 312 ·914 144·

_ ~-----------------·----·---····-----------Z.8.!L ____________ .l.ll;__ ------!U:.. ____________ 1H......... .a.aJ,___ -2-U----- - 9-3-S- ..... 0----- 1•1 - IH-8 192 836 240 886 216 936 2l6 916 Bil

__ lU ____________ .8t7~---~----1t2 ____ --.&i1---~--------i---------9-U--- -----~~~l-4-0--'----- -- -~---O-t1:--~----~-- s.ta·~--88 110 &38 168 888 S44 938 192 988 0

--~-------~--------~---------- - __ ,_89 ___ ~-- -- ··- !..84 ___________ j_3_9 ____________ ~.Jl- ----------··· ~--· -112- .. ----- -9-19 . --~·-···· -- ···---1.01: ..... -- .. ---t-1-9- --····· ...... 431• 195 192• 840 96 890 288 940 48 995 432 79{ .. _____ Q ... Ut 12~'-'·- . 1-9~ 2l6 . 94!. · -Jtt· tti · ó 192 110 142 312 802 8 942 t44· 992· 96 193 9.6 . ~ ~--- ~- 143 144 8-0-J..: ,,. - - 943 - . . 0- 993 144· 194 5b4 144 12 894 136 044 '2 994 101' 1.9~ .96 &4! 360 ~ lt-1 0 . 94!1 3-34- "'''' 19-1 796 0 846 480 806 48 946 1012 996 144

- .l-9.1--- lAG. - .U1 0- 8-01- lJ2 · 94-!7 tiG 09-'1 5,;68 198 192' 648 72 808 l44 948 144 998 3!2' 72.9 --· 0 - 149 3.16 -8t.J 4136 949 144 ,,, 0' 8Ö Ö 84· 150 240 9Ö ë tJO 9JO !i'J4· 10 Ó Ö 144·

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Page 8: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

-----------------------------···--·--··-·--···---------------------------------------------

___________ _tt ___________ ~U~~t ____________ lL ____ HUl!.. N . ____ JllJJ!l___ ___JL ______________ Jtl.IJ4---------- - --tl----- --- NUM"

1251 stto tiD1 6oo _ lJ!"-~--- ~-4_0.1 _________ 316 ______ 1411.:____ :.u.t· ---- ta5! ---o-6~- --1io-a-·----- -i,,-2-- t:tia· 21o 1cto2 1•1 t4!12 1o4 __ i2!S3 ----~84_:___ _______ 1803- ________ Jt _____ tHJ~---- ___ ttt ______ 1_~6.3 _____ :U_i ______ J._4_U__ 16l

1254 288- 1104 264 13.4· 264 1464 0 1414 '10 ______________ 1255 ______________ o___ --~--- -~.BJL iü.t· ______ aa.a _______ t.fl!.: _: ___ -----"'-·----- .uu.· -t-

1256 312: 1t06 216 i!!J6 i44 1456 528: 1416 48 _______________ : ____________________ i~;:- tt!~------rnt -~•g- r::::__ --::t-- ·i:~~- mt i::·r ~:-

1259 ~~l- ______ 110i . .118 - -- tl.H - .J!. 1oid9' 411~ -1419-- 2-64· ------------ -----1260' 144· 1Î10 432 1!6Ó 48 14i.O 384· 146Ö 214Ó

__ 2~-------- 1! S.1 ------ ... JL ----1-~ _ .. l..ltL-- .. 1.4!1-:- --- - .. 16"-- -------- --146-i: -- -- - U-6· 1242 432 11:5.2 48 t362 181 14i.2 10t· 1462 191·

___________ -----------------------------il~J- _______ _0 1313 _:_ _______ llL _________ .l!.U~- .... .!.~L--------Ui!... --- - -----3.6..0---- -----14-U~-- -- --0 1244 0 1~14 576 i!~4- J36 14~4 336 1464· 144 '

------ ___________ ..i2_H_ __________ _j.8JL __ -1.U_!_ ________ 1,_'-A:.______ i 3 '!. . ---------S--12----- 14' !5 :_ __ - -----------O:-'-----------t4U---------l0-t-'---- -- . 1266 384- 1816 218 1St6 116 14t6 111· 1466 696·

----·------ ---------------------------·- -··-------- ... 1J.6__7 __ --···-- ..... tiJ.:._____ 1817 ________ _aj.Q__ ______________ ,~~,- ----- ---···. ---c-0-------- · ---14i1 .. ----- -- --1-t-1:... -···- -44U -- t-ftL ..... 1268 120 1118 120 !318 216 1418 408 1468 0

_________ t,!..69 _____________ ~~4..,_ _______ 1Jll..._____ __ ____ jl __________ J.3•• --------~221 .tAU .ua, -1A-6t -'-'•~ ----·-1215 240 112o '' ts~o - t28 141o '' 1476 411· 1!2L Q n• .. i.IU... ..aas . 1311:..--- .... 2&8. .. 14.21- 64..8 14,1 G 1272 1~4- 1822 !504 1312 8 1412 218 1412 ó

.. it1J ---2-'! 1h-3 U-4 - iij-J. - --216-- -1-4-13-- - ---0--- 14'-3 3-3-6-Ï274 5~4- 1324 '12 13Î4 136 1414 !44· 1414 192

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. 12.1..1 ___ .lt _____ -UH. .43.2. - ----1-tit. . JU - - ... Ut--9...-------------~ !41-9--' - -Ö-- -i2eö 24 tilo 21e !sao '' 1430 384· 148à 144· USi· 2tU ---- Ul1 288. i3t1 112- - - U-11- .------~- !-411· 614· i282 144· 1832 120 1312: 4!56 1432 72' 1482 218 "' ' .· ,, """ ' . - -- -- --- --t~:-!----- --::--:~- -- --tl~ --- -· ~:: lfft: sa:-- ·-- ----t::: · ;~- i::: i::,

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__ . _______ _____ ___________ _ ___ ..12_8_~ __ . -----··--- ______ Jl~-----------UD ... ________ .Jaa ____ -------1--tU:------- ___ J6-._____ -- --1-4-'''----- .... ----2-U------- -- .. ---t-4-U~- ---- -t" 128& ,,, 183& a•o i3&8 aio 1418 1ta· t48a 48,

.1.2.8..9 ___ . ____ 4_sa.L... _ .113t _______ --112. _ __ ... i11t 13-6. __ --1410--- - . .. o t41o. ·a•o Î29Ö 192' 11-0 0 t300 248 1440 110 140Ó 4~1:

.... 1.291 .. .2..1.6.-... UA1- - -----"-'-- ... Wt~ .. - --8--- -l.-4--4-1: - .. -U·; 140i 28-8-1292 96- 1842 240 1302 D 1442 288 1402 120

j.29..3 2as .1!43 o. . _ iJ-tJ.: so.a -. -1443- &o.a· t4o:s 264· 1294 !36- 1844 48 i39.4 116 1444 1!6 1404 6óO 1.L~' _0 1.34..5-- .. .192 i3t!: 218 1445 4J6- !405 0 1296 1~2: 1t46 336 t39.6 S68 1446 314- 1406 336

__________ 1ll.1_ ___________ J .... 4.4:. ________ . .....11.!7 ___ ~--- .. i4Á . - -1-P-'------- . -IAJ - 1-4-47- - .. Ó-' . iU1 240 i298 384- 1848 06 !308 . 148 1448 2l6 1408 101' llH__ _..J.t.t~- .1U.i___ ''-2.~ tuo o t-4-49 .• ,,, -14t9- 3l2! i355 t65· 1''o 48o t4öo z4o 141o 1'a' 1soa o

---------- ---------------- .. --- . ----------------------------------------------------------------··· --------------------------· -·---·- ----~----------------·---------·--

Page 9: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

.N _ .... . IUM ... NUM

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-1-52!. -·--- »-~ -ll11·--- •.o-a. !•»~ -ti~-- -1.6'11 ······-····-----o·--- --t,tt·· 614 1522 240 1112 144 1612 360 16'12 96 1?12 218 11~23- 16-t· 1J7-3 26-4 tt-i-!: . it 16-f3 38tt: 1'113' 110 1!524 240 11?4 648 1614 t92 16'14 ,,, 1,.. 0 .t52J 3-'~- 1J?-5 ..... ~ iûJ ,,_,,_ 16.!1!5 16! ........... 1,_,,, - 48(t· 1!26 ,,6, ,.,,, 120 1616 240 1676 216 1'16 288 HJ2'! JL ___ ill_1_ __336 . _ i6.t1-'- 1-68 16.'7 l92 !.11'1 0 1!528 o&. tt?8 192 t6t8 e . 16'18 240 111a 31: i~2.9 '-~~-- J.Jl~ 2.1 . .6__ .1.U.t_ .6Q.Q_ ---- 1619 - -0. 1'1.t. - 211

J:~t ::_:: ____ it:t 1:~ .. l::t "~t . i::t .• :: .i;~~. ::i 1532 0 1182 112 !632 48 1682 371 1'132 144

_ __ __ 1,_3!1. _____ 2_8_.8 _______ ttl_:t ____ -------4--------t-61!-- -- .. soa ---1613- -24-0 -1113-:___ 40& {!34 240 1184 12 1634 ,,. 1614 110 1?34 486· iS.U. __ .. UJS.3_ ... 2l1t .i6-U !J2 16-.U 6-14.· - -1'11-J o il36 ~•· 1'1' sae i6~6 soa 1616 sas 1?36 288 i!S.3.'l...... .. -.19 .. 2.:_ ..... .tl.a7 ........ --.-:.2-0.Ó.---------iti-1.:. ------------4-U:... ........ --161-1 -----······.. ··---~----------!1J-t.... ..:@4-ó {S38 SI! 1188 ?2 i6S8 432 1618 110 1,,. 191

____ il3i __ 4tLi·-·--·---.1•lt___ .621l.: ....... ------------i6il:.------------·---·-Ó.------------16-I-t---·- ·····---432;---- -1-,39----- --4-IO 1546 06 1'9o 314 !64é toa t&tö 311 17~o 96 15.41- Ut: 119-1. -0 164!· 118 !l601 431 1'14t 3tS '1542 2!40· 1192 240 t642 t:68 16t2· 0 ·1'142 !J!I 1!43 o, t-193- -- t92- - !6•3 24o t-603 - a•• 11•s· o Ï!S44 2!40 1J94 408 1644 !44 1604 624 17~4 ?I t!54!5 __ . ___ 2&! - s.t9s 384- -· !.6~-' -- toa 16-t! o - 1'~' ••o i546 4b! 1196 0 t646 128 1696 '2 1746 ?20

-- il4; - -U-1" 1Jf1 168 t&i? o 169'1 :SI-6· 11•'1· 1tti 1!48 1!0 1 98 384 1648 0 1608 102 1748 1140 i!49 21 & . 1199- 0 !6~9 ,,, 1-61-9 264· . 1 ,,, 4&0 Î!S5Ö 6'2: !100 30 16~0 480 1'100 3S6 17!6 288

Page 10: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

-----·~------·-·~··----~- -·----·-·-·-··--------~--~- NUM

_ ·--·- ~o ______ __i_8JJ 1.:.. _ _ ....... 3.3.6... . .... it!l_ ... .... . ......... 113 e. ___ _ _ 19U~-----··-- ... .lU·. . ..... UJ.t: ---··--· .. -ei -- . ,,, 1~02 314 ia~a o t952 432 s.o1a· 12ó

.z~_o __ _ ___ taJl3._ ..... 1t.a. iu:s .. •!.2. . .. 1û.3 . __ o. . __ .. t.tJ.s: ••~-11~4 486· 1804 144 tli4 ,.8 1904 0 1914 336

.... i-13.! .. 18.4· . .. U.Q!_ !0.4. ilf.!· . -0 1-9~5 21-8: · tO.fJ' · 218· 1156 0 1806 480 ~.,, 72 1906 140 1916· 240 jJ~7 . --~~~·--- ___ S._U_l .. 0 1U7 24Jt 1907 311: .19.U:_ 102.: ·------------·· -------------- ---------------1'ifi 214o 1ao8 '' taea a4e 19o8 2t• 19ta :sa4 .

. i.!~ i .a .. 1.11lt .. -·-··· _ !'l6 ..... __ .!U$. 368_ ... 19Ö9 336: . -1U9c. o Ï16Ö 1144 1110 240 1860 292 19l0 ,,. 1960 168

__ ll .. 6.L .2c~------ _illL--.- 5.12.. . ~U: .. A!.6 19U -.0- .. ---U-6.1-' Sl4,. 1162 aas. 1112 144 1862 !ö4 1912 t6 1964 36ö

_ ----------------------·-~-· _______________ 11~_3___ . ______ 2 8 '----~-__111_3..:._ ___ .__21.6 ________ iA6L ______________ a_ ______ 1JÎ! --- ---- .... 4.!t: .... ---·· --1~-- 144·--i164 270 1i14 512 {864 06 1914 ·!16· 1964 216 i 165 _______ 2!4 G -----~--J.W.~-----------O ____________ ilÁI~----------5.:7.A_ _______ 1ti3 ________ .141 .. _ . ---..U'-!l----- 48.Ö-

-····---·--··-··-·--···----~------.----ff6f- söo. 1116 168 i866 384· 19t6 ó 1066 •st·· ••·fl!{· ••••~<<•<<9<~:~-m•---·---i:iä-----«m<<<••••i-:~•<<<,<<< i~!!: < !!:. i:f: ~:t • •mi::: 4::

. ~--~-~-~--------------------------~1~! -------...... l-~l~-------~.1ll..~------------.2..4Q_____ ____ iJAI _____ . .'-14----····-- .1.9{.9 .... --0'- ....... 1.Q6.0··· 2-ó 1110 480 182o o ie~ó t91 1910 48 1976 614 t1'li tl2~- .. .1U1 - 43.2 -- i8,2.~ -0 - 1011- -- 444) - -1-0-71: - ··- 384 1112 110 1122 102 ie;a 110 1012 306 1012 144 -i17~ 311 1113 . -0 iajJ 144 . 1913 IAO 1913 1&4· 1714 2~0 1124 96 ~8,4- 612 1914 101 19'4 384·

_ "_,.7.! a .-l.UL __ Jl6 __ u;..J Jt6 to1s -•'-a: 197!1 o i''' o 1126 672 te'' soa !916 3&o 1''' 336 i7_1; 2Jt!t 1U.7 288 .. iti7· .t.óa ... .1la7 o 1011 411· 1178 480. 1128 06 ~·t· 288 1918 240 1078 144 1.17!- 2~0 U29 . 480 1t't . t 1919 314· . 19-10 I!JI 11so ••· 1a3o 288 ie•o 24e 1930 a•a- 1016 o

.. i~BL 516 __ .... 1131: ... . 0 .!881: .. '12t 1031 -804· -·1-911 24Ö i782 408· 1832 312 1882 216 1032 06· 1912 218

·- }~~-~---------3at-----·-i-!~!-----------!::c -f~f:" ~n!-- ···i:::··· --- - :~:. -t::: Q . . Q ___ i.Le.l_ . ________ 9.!:... 1A.H ___ ··- .. 240 ia&!:. ~~2- - 10-SS 0 1011 480

---·--- - .. 1786 2140 1ê36 102 t$À6 ,,. 1936 78 1086 ''' -·---------------------------·------- ________ 17Jt1 .. --·· ____ 16.!~---· .. __ 1J.!:'L __________ 2.81L _______ iili.~--- ---~--.ä-~-~---1t3-'--~--- ----5-!fl., ------1U'1-'- -- . . S,61

1788 o 1l!e 336 ieea 12 1938 384 1918 2ea ·--·----······-------·······---·-j.18!. -----~!1.2' _____ 18.3.9 .. ~----- .... Q _____ AB"--~--- J6tt.. .19:! - -111--- -----1tl0'- 11-6-·

1190 !24· 1140 48 1800 884 1040 240 1900 288 1'191 o 1141· 48o ttot· . a•o. toet aaa, tooi o 1?92 0 1142 192 1102 144 1942 264· 1902 06

.. i193 ~:u. . 1143 144 _ Ito3 a•o 1oc3 . ó -1tt:s 288 1194 384· 1844 360 ~·~· 360 1944 3tl 1994 648

.1?2.' 1~2: ... 1.t*!L_. . .. 4&o. _ . 1aos . o 194! t•a· 1tt1 tora· 1796 240 1146 336 i806 240 1946 8!6 1996 '2 i1.2.l 288 U~1 0 li.Ó'1. i02 1-94' 101· 1907 !S04 1?98 2~0 1848 06 1808 336 1948 0 1998 ,,,

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Page 11: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

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Page 12: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

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. _22?.5 ______ 2~_0 _______ U.H~---·---U-6 2.!,!1· .241!- 336 ........ -247J ·---- --36~--2276 184· 2~26 264 23,. 384 2426 648 2476 12ó

... .227._7 _____ a_.4..Q. _________ _f!.2.7 .. _____ - .... o ---... IÜ'---- -----t-U:-~----2-4-t?. - . --336- -----~-4~1------·-----4•6-22?8 ëSB, 212e 192 as+a J!o 2418 o 2418 3&4· 227-! .... Ik - .a-t-a• 384 ai?•. 3-14 a•a•- .,,. ,.._,_,_ 6· 2280 192: 2130 720 2310 96 2430 614 248Ö ,,

___ 228t ________ a.A.t._ ......... 2SJ1 . . 410. .2Ü1-- -·- - 416 2431- -0-- - ----2-411: . 6!!!4· 2282 480 2132 0 2382 336 2432 '-2 2482 288

- __ 22_8_!_ _______ ~i_, _______ J!33 .. 408.. ---23-1.!.----- - . Q.. 2433 240-------- -2413 480· 2284 120 2834 128 23À4 i68 2434 384 2484 384-

. 2-au.. -n.o atu~- --{) 21-it· ••-e a.,, ·BI& 2481 18-& 2286 ,,, 2~36 192 2386 384 2436 192' 2416 671'

-----------228'1---- 0---····· -2131-----···----·3-8·4-----·---2-Á-'1:-·-·· -211- ---2-3-7----·----2-16-~----·- -2487 0 2288 ó as:~e 288 ~~•• t44 2418 336 2488 !A4

.... 2289 !8 4 21.!9 . -----411-6 - -- 238fc 4-08- . . -- ~4-19- --- - ..... a 2489' !128 ~295 24o as4o aas ,3•o ,.,6 2••o 144· 24to 384· 229t 4.32: 21•1 ~•o - t~tt· a 24•U: t12 2491 rus 2202 192: 2842 504 2302 ,, 2442 314 2492 ó

- _.22.9! 169· - __ 2t!'$.3_c______ .o 1303: - '-2-0 --- --2-443- -- 144 2403 369 2204 912' 2844 216 2304 ?20 2444 240 2494 240 220!S 0 -~14-'----- _6~l_2 230!1~ .l.!!. __ ··-··- .. 24.4!5 . 384 _ 249!L _rj _ 2296 t) 2: 21•U 288 a3t6 e 2446 :ss6 2496 o ~2-~'L 480 ___ ._2!4'1 120 . 1317 114 2447 0 249'Î 336 220e 384· 2S48 168 23tl a4o 2448 144 24t8 480 2299 _____ ~_.2_6_4 _______ 21.49· 648__ 2.3t.J ___ ~--·----~-O-- ---2449 .... 410.. . -24'-0 .. - . 4i'U 23ÖÖ 0 2i!O 410 24ÖO SlO 2410 ''6 2BOÖ l'D

Page 13: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

. N.UM.

asiH ''2: 2,5_1 __ ____ o t6o1 110 26.!11 624-.. . 11t11 !5'16 --~,o2____ sd4 ii52 e•o t6öa 312 26!12 o a'o2 ,,,

--------·---------------------------------~;%-l -.3~ ~lll ~~l- ~1BI ·i:: :::~ ;,:_ ~~~!--- ,~---~'Q' _ . 48 o 2'!HJ5 28tJ !6Q! t9.2 26!.! . o 2'10!5 864· 2SQ6 .ä4· it., 0 26Q6 228 26!16 12Ó 2106 384 2!501 336· 2f5' 2.l~. 26.0_'1 ___ !. 2.6!1 314 2'10'1 _ 1JI

-- i!soi - . o~. -il58-- - 168 26Ö8 24 2618 288 2'108 360 a!Soo 1:!'·----- atlo ____ o .aecnt. 1aoa._ __a,.Jo_ a.s.a: a'o-o- 4&.o 2!iir "!4· 2!J60 24 26ÎO 1.20 2660 218• 21tÖ 218

.. ar~11 __ o.· 2!~1.:..... . _____ 1_68 ________ .AA.i1 sta_ au1 .. s,, -1-,11' o 2812 '2: 2962 288 !6i2 !68 2662 218 2'112 240

_________________________ .2..5.13 _________ 18 ~~-------JU;L ________ --1~.4.. 2Ai!: __ ........ --2-18-- - ---2643.: ----- .. 2'1-J: .. __ - a.a.a. 2514 480 2964 336 t6i4 !552 2664- 360 2114 960

___ i!~!----------1.4.1~------HJ.S _________ _z,_a_ __ -- --------.16-t!i·---------··- ~- ------ -- ~«~- .. 314· -2 'ï-!i-- ------ - -- 1 'I.· ... 2!516 432: 2!J66 216 26{6 336 2666 8t6 2'116 0

... a.a.t-1 .aa.s, .. - --2-J-6.1. ---.0. -16-i-'-~ 314- - 266'1 t-tl a't' 584 25ia 3&ó :1:: 102 l&~e 480 2668 _o,. 2111 4!2 ________________________________________________ nu ________________ ()_:_______ -----------31.4" ---t«-9·~----------a••----·- 16-6'-' ------- ,_.. ____ -----a.,t-9 o 2525 2~Ó 2110 528 26~Ó 0 26'10 480 1'11Ö !44· -3521----- -384'-----~lt71---------H6~-- ---26H.c ---'144--- -26'11 ····-lt·-- -- t'ti ·614· 2s2a 412· 2112 12 26~2 •sa 2672 o 2122· 240 . asaJ a-!- at7.:s. sa& e.&f3 .a 261'3 614· 2113 211 2!524 0· 2,74 120 2614 96 26'14 384 2'114 240

. 252! 840_ . 2J75 0 - Ui! !U6 24'5 J04· --- 2115 360, 2!26 720 2176 192 26.6 432 26'16 144 2'116 8l6

. 2527 _ _ a ___ 21!7 _ .. a4o _ a6t'1 aas .a''' a••· '''' o 2!28 ,,, 21'18 192 26è8 240 26,8 480 2,18 144 2529 7 .. an . tJ79 !04 .. tti• au a''' o t'1to t6o t53Q aas at8o t92 t&3o 720 2680 t44- 213ó 384

. 2531 _4ti$ a, at 43a ---- ••~1 ---· --- - -o 2111 s1• 2'11· - 164 2532 240 2182 600 2132 96 2612 360 2'132 12~

___ 253Jc. 19.2.;_ _ __ 2JJ3.. IJ-33 -18-4 · 2683- 110 · 2-1-J:S ·- 38ft· 2!534 720 2184 192 26.4 432 2684 D 2'134 336

..... ..H.3J.. , _____ JL . 2.1.1!_, ___ . . 48(L- -1~!. 2-81 261!1 218 2?35 -ó 2536 165- 2t86 6'12 2636 264 2686 384 2'136 120

_____ 2.5.3i .. _______ l!Q;__ ... 218.7. ---S..92-----------I-6t7--- - -- -6-41 -- ---- -161'1 - -- ···· 0· -213-Y-- ---191• 2!538. 384· 2188 Ö 26.8 240 2618 48; 2'S8 468

________ ~- ..25.:Ht ~- ·----26 . .4.: _____ --· .!J.SJL __ ,____ -384,-- . ~6S9-'--- - -- - --o-- - --·· --2-689 -- -- -288· 2739 - 384··~ 2540 2-~ ~t90 288 t&4ó 96 2690 ?20 2'140 1~4 2.541- 5..2! 11-91 0 .. ,,.ï. 248 2601 !,,, 2'14!> 606· 2542 1~2 2,92 108 2642 614 2602 144 2'11t2 384-2543 Qc.. 2f93 -- -- - 288 2643 240 260! !04• 2'14~ 0 2544 o ~~94 s-ae t6A4 216 2604 120 21tt4 384 254! a es ttts- - . - aaa 2,_., 112 a&O!I· o· 2'14!· 8'16 2546 769 2196 240 1646 6'12 2696 288 2'1tt6 4!6 2!4i .liO: 2,97 648 !6.f'1 B 260'1 tt8 -lf'' - 4:JI· 2548 168 2!J98 288 2648 264 2608 288 2'148 0

.H49 ___ !40- 2899.- -- ~ -0- t6A9 sas 2609- - !-60 2'49 !04-· 25'Ö 336- 2800 360 2680 160 27ÖO 214- 2'1Ö 864·

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NU'" __ ~~ _a_ -~~--~~t•o~1 ~- ~120~ ~ ta!t_ _26A 2oi1-:- __ _Jt-t'- _____ ~-20!-i.:~-~- --~

~1 24· là02 288 18Î2 288 2902 2!6 2912 2-0 --~--~--~---~---~---- --~---- -~~ . ~ ___________ 2.2..53__ _ __ 33_6-___ _ .JAQ.l__ 216. -- - --2.13-- --- .360 -- ~UÖ3 ti- - --~--19-•3-: ~ 28&:

21~4 9!§ 2104 408 28;4 312 29Ö4 264- 2014 060 ... .2J!.!!J 192: 2.!05 .576 ,.,, 0 29~J 288 2915' 181

27~6 480 2106 432 28,6 288 20g6 ''2 29,~ 12Ó ___ 2.!~-'l _ '~'-- _ 280 'L a 2817 24! ~ 210'1 !88 2-011~ - 641· -

2158 281- 2108 192 2&ia 416 2908 0 2918 384 -- ___ 275.9 0~ ~ - 110_9_ - .318 ··-··· .1&19 -~ -432 29Öi 840 2.019 0

276Q 192' 2810 480 2860. 96 29:[0 4ao 296Q 19! -~ ~---~tl.6.1. 131.: 2.111.:~--- 3.1~ !.8,1-: lt-2 ttit. . 0 ... 2.061 •••.

2112 !12' 2112 0 2862 .,, 29!2 144 2062 191 -~-------~-~~- ---~---~----~~~-~?~_3__ _ ___ 3_!_L _______ f_t1.__3~----~ ... 12 .. 0.... -~---- 2S6l ______ -~-JL . .2t.i3.. 336. .... - .29U~ . 3!.1-

2164 120 2814 480 2164· S20 29~4 288 2964 288 --------~-~--------------------~- -----~-------~---2 '-''-----~-----~l:7 2: _ 211' ~--------n. _____ ~------l~-~-----~-:t.a_._ -------"1.! _ ---------''-'-;___ ... ao•s~---~--~ .-a•o -

2?66 336- 2116 24 ta66 •ao 2916 318 2966 !032 ___ 21J.t .. tv___ 2!11 ..... ~-------- ___ .. 41.tL. all-'. .a.aa ~~~' a•tt- 2961 o

2768 169 2èl8 2~0 ~·~· !92 . 29[8 408· 2968 96 . 48 2 9 ._.._: --' ~---- -~ ,. ------------------- ------------------i-l~l- · 43~:-------i:la·------ -:~-:-- :a*~- -------- .~t ----::i: 1-:. ---::;~ ;::·-

atzi_ _,'t~-~--- __ l_tl1.-. __________ 31~-- _____ ._.,1. . JL .2ta1 ____ _ ___ ,,_. _ at't 264 2772 288 2122 576 28ft t44 2912 518 2912 0

.. 2.,;73.. 192: 2823 .. 0 11;3. 624 2113 .. 144- 19'13 336, 2174 672' 2124 288 lé'4 410 2914 288 2974 '''

... 2775 _Q _J.è23 .. 672 l!il !88 2015- 1-40 2071 0 ~{76 110' 2126 936 28;6 8 2926 48ó 2916 144 2'1.'1'1 . 410· . '"'- - 192 28'17 4.80 __ ,.,., . 0 2017 288 2778 4~2 2128 144 28+8 s92 2918 o 2978 12ó 211.9 . 3.36, 2!1-t 384 "~' 0 2-919-- -41è. ''~"" 360 2?8Ö 0 2830 240 2810 72 2930 336 2915 192'

--- 2'l.8i - . 5~6; .2!31 -- 0 28.1 -- --336 1031 336 29as. 960 2782 33·· 2132 48 2882 768 2932 168 2982 288

______ IJ'l~---- __ o-_ ________ ~2A~3 ________ _3JI_~ ________ !ll:S.:______ .. 2.48____ .. 2.9.33 . .!1.8 -- 2913 - - - o 2784 1~4· 2134 1008 28.4 $92 2934 792 2914 3ll'

_.!l.S..! ___ .. ____________ u_4_, _________ ws _______ 43.2 ... ___ .a.aaJ.,__ . 4!t.. ---tt--ss --0 · 2011- 384-2786 864 2836 120 2816 480 2936 480 2986 360

_________________ _ __________________________________ _azai _______________ 1.!.4.: ____ ~-----2.t3"l ___ ---------'1.2-- ---- -~~-------------O---------.a.9s-1:.._______ ------U&------- -2N-1------------------48-o--21ss 96· 21!8 410 2818 228 2938 288 2988 216

.21 8~ -· 9 8.4. ... . 2t 39 --- ---· -----0------ -2-8-l'-· --- ···~--- -------1-9-39--- - . 696----------- . !03-9'-- - - lo-4~-.... 2i9o · 480 _____ - al•o ··- 3a4 2s9o 4116 2940 o 2ooo -tso 2.29~ 0 . . 21•1 JIEL 28-CIS.- 104- 1941!. 336, 299Ï' Ó 2192 3!2' 2842 216 2892 96 2942 48ó 2992 4a

.. 2?93_ .. 4:!2; _ .. __ 2j43 360 2-8.3- lOl 29413- ····-- 0 2993 1'16 2?94 289· 2144 o 2894 116 t9414 _.a. 2994 ,, •. 2Z.9!t__ _ 2~$- . _284_,_ :!84 t8tl~ o 294!J· - 74a 299Jé 102 2196 2-0 2846 a64 2896 S28 29416 314 2996 3a4 .21 .. 9J'. 21.6· 2.&~7- . 0 . ttt1 4!2 29-4&'1 102 ,,,,, ·~·-2198 !as 2148 96 !soa 41B 2948 2•o 2098 216·

___ 2.1.9_9_ .o ___ 2.149 768 aa.tt- -••o- 2940- :s•• aoo9 o 2a6ö o 281o 48o 2tBó 368 2910 Só4 3oó5 28a

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N NUM ·~···' "-"- ----·~·-·' N.

.... 384~ 288 .. ,., 412: 218· 1tà

0 144 060 384 :S6ó 240

"'' !ltS.

Page 16: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

---·-~·--·--· -- ---------~ -----------·------·-------·-·---··---------·--·-·----···---·--·-----------·----····-·--·----···-··--------------·-···--·-·---------·-·

.. . .. - __ N__- .. JiU.M .. .. N. NUM:

?'44· !8 01 4 0 8 3:SÎ1 3401 ___ i1Jt~~- --- 3_411_ -- - - 2!8 - üu.- ----·ssoa·--------,,-a --- :fii2- - - t68 i•o2- 614 341a o

-- ;~:-· -----H~~~--- -- -14~ - ~ft} - ~'-: ·l:t~ -·- -;:r-- ----~:!: -~::-0 tlQ_, ~~1(, 3Jb . !,2 34.Ö5 5.,.. 341.!· 0

i~i- 3~06 516 33Î6 0 3456 !36 3416 96 4~t _ ~S_Q_7 .. 216 33.t't _ _ 'Do_ :1407 ó 34!11 _ :u6,

3258 641 3-08 168 3318 314 3408 96 3418 480 __ _ -~2_!'9 ---· _U.6_____ --~_to_t ____________ ,J,_ _ .ll!_t_ __tL :1459 ''' _ .. 3.41.9 __ 288

326Ö a !Sla 432 3~~0 96 34ÎO 410 !466 19!· _____ H61. . 5.!1: .... .!tu.:.... JL :!h1: au.. !4i1- -34~- ---346-1 11!&-

:1262 384· !Sl2 o :ss&a 491 3412 120 3462 38~ ____________ ___ ______ _________________ _______ :12_63 _ .. __________ 0_ ________ 311.3______ _ _ _.2_1UL__ _ ____ .3.3J3 .. _____ JA4. _________ ... !4i3 .. ............ .. !l.á-4'------ ---- !46.~- .. - .. . . -... 1} ·

3264 4&- 3~14 1oo8 :~364 2'4 3414 11e 3464 saa ---·----------·-·-------·----·---·-----· ----·-----------H.6_!t __ ......... __ _L$! 3 ~ 15~-----tl.2.. ________ .31AL --- --------lU--- -- ---3.Û5- -- ---- - --Î-------- 3 4 6 S ------- -44-Ó---- ---------

3266 864- !Sl6 264 !366 1108 3416 128 !466 312·

--~~-~j--- . -:~t-- ~ltJ------- ·ti·t·-- ·ft!~····· J1~ ~:fi ... ;!t -~::-: t:: -------------------·------··------ ------1:-~à---··-------~~:: ______ ~-i~t~~-- -------24~---------:lro··.------· ---··::~- ---- --~:i:------ -- '':: ~:;~····· ----- ·:::.---

- .1~.71 ...... _________ 0 ___ ltl1 ____________ t6~--··· ··----3~..1,1 ..... 10.4... . ....... 3 . .4.11........ ..!118___ !.471· 0 3272 336 !S22 192 !3t2· t~4 3412 614· 3472 ''

. 12?.3.... 432: .!123 41l8 !!i! 312 .3413.. 0 !41.3. 480' 3274 360 3*24 o !3'4 864 3414 12 !474 '2o

__ :S.2'l!l . ~00 lSI5 . _ 336 . _ !!i!: Jl !U!L 6'2·- :S41B . 360 :1276 24Ó 3.26 ,,, :s:s'' 72 !416 1'6 !4'16 384

~.i~-~- -:~t ~!:i a: :~ti -:i: ~!:i- i:!. --- ~:;: :::· 3279... a .. !ia9 . 112 33;9 a•• 34tt '''-O :S41t o 3286 9S !$30 720 :s~8o 360 3430 314 :S48Ö 101· .3.28.L_ 960 - _3.~!1- 36Ó !31.1 612 .3431- 0 - - 3485: 366 saea 480 !S32 !36 3~82 240 34!2 1tt !48t ,, •.

---···---- _______ ------···-----···--·--~--·-·--- ....... !.!.8.3 ________ ...2.L&, ___________ Jt:S3~- -U4 --- -J-S.J.:. __________ --0 ··· 34SJ..- 28-1- ---- · ·· -- 3483 - -- 408

----········- mt ___ --------~:~ _______ :~~:_____ 360 .ll!:L ::: ::::. . :::: ~::;. 8,~. :si86 6@4· 3~36 192 !3.6 192 !436 168 3486 116

---··----~-------·-----------~---··· ~---------------3.2.8..7 ........ - --- -Îl-i. --------!U-' ------------a••. ····- ... -- --33-A-7--------2-8-8--- ------34-:J-1 ... ------- --·3-8-··- --- - ---!4-8--7------- ----- ----(l-

-----·-----·-··--·····--······--··--···----····-····-------------····-·--·····---·--······--'3" ... 2 ...... 8 ... 8., ....... - _____ :; :. ______ Jt~=------------llt. -----······ .:::: -------------·~: ........... 1::! .... , __ --- --~-~~-·-···· ··-·· ..... ~::: --- .. . i~:-... -3290 168 3~40 144 !300 384 3440 0 :S49Ö 240

. ~~~~ a•r ~l!i . :t:·' :~-:i- ,: ;::i ::~ ~==~· ::r . .3.2.9.3 - 52! - -· ~~·3- ----- ... 0 - 3301- ,,, .. - 3443- -- 3-84+- - ---3493 384 3294 ''~ !S44 240 3304 624 3444 288 3494 9!6 32..9! . __ Q.e. :SS4J· 480. -33-0S' . 480 3445 288: 340! à 3296 240 :st•• 48o !306 336 3446 606 3406 240

.. 3.2.2.1... .. 48D .3Sf2 264 :sso' a4a 344'1 o 3407 384· 3298 192· 3S48 192 3!08 696 3448 06 !498 480 :.t299 t4.l. :Ji49__ .336 :s.:s.tv o 344g iaóo 3409 264

- 33o'Q' 33 6 3~!10 840 34Ö Ö 240 !410 6'2 !!'JO Ó 288

Page 17: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

.N.Ul!t.~- NUM·

··············-~---·····--~--~--~--·······~-~------~-----·-····--------~~Q1. --- -- 7.~2.' .. ---'~'1- a. . -~6§1: 410 3611 432 .. 3'101 ····· ..... -1080 350~ 1'2' !~12: 144 3602' §76 36!2 144· 37Ö2 518

-----------------~----~- ------- --- ------i-~~ 9~ lil~ - ::~ ~t~r ~~= ~~=: -·:!:- --- -~~~t 48~

~~~~ :1:: ~11: i-:~ ~:l: ~:: ~::! 4!~· ~;~: ::: 35Q7 192: !f51.. 4Qél .. !AÖJ.. . -0 3611 410 3'101 3!6· 3508 120 Sfl8 336 3651 336 3618 ·288 3708 d

_ ~,_Ö.9 JC)i~ S!J_9____ _Q_ ..!Ant. t!O - :!619 616- 310(L 1.04-· ---!510 ''s ,,,o 28a S6ib 4'' 3660 1ta· 311ö 6?1: _3,11 .... l 3.1111. .. 91 .. 2. !6.!.1 624 !661 s:u.. 3711 0 !512 240 St62 336 !6Ï2 D 3662 614· 3112 24·

...... :!!.13.. ___ .4.\li.: _____ ~_lli!:._~------ .. !J.8 ____________ .J.A.i.S.: ...... _.~~Ua---- --~------.3.663--~---------0-- ··--------------3'-l~-"- - .... - ---'61--3514 432' 3t64· 216 36t4 120 3664 110 31t4 ,,,

.. ------~-------~-~-------- ------··------1!.1.!.-----------'An-'-~-~---U-'-5 --~-------281l ..... -- -----...1~--~--------.f- -----------1-UJ -···-···· ----- --5-·6~------------ ····-!' 1-s-~- - --~--- 3-3-6, --3516 0 3166 1200 36Ï6 96 3666 5'6 3'116 431 3.!.1'1. .. . . 3.6.Q.. . ..!J1l7 --.0 . 36-t' .. 'JIO- 346-1- 249 - !71' ,!-0 ssîe ''&· 3168 o s6ïs ?68 3668 a•o ,,18 111

----------·------Ui-9 . - -- ----- ! -~------3fA'--------· -8.1-6-~-------Ut-t------·······--···-----11~----------- -- · -366~ --5,6·------ ---3,,_19 ----o-3!520 ö ,,~0 !84 !6fÖ 288 36,0 431 3110 288 352t-- .. 612" -3-1-11~-- -36-0 -!61-1 -41-o ----3!-'1-1·--··---- '0 3'11t· :166 1

3522 384 31?2 336 !612 312 '''2 192 '''2 648 3523 t•-4 Jt13- 360 !6f3 · · B 3-6-13 3·36· 3113 288 3524 480 3914 456 ,,,, 336 3614 i086· 3714 161: 3!52!5 - 48.tl :SJI7!5 -0 ---- -l-6tJ . _,,, 3-6'1!5 - . . lf-1: 3-,., 8-40 3526 ''' 3176 336 ,,i, 1110 3676 0 3716 816· 3527 o -- -311'1 -- t36 - s6-ê' --2-Ae 36'17 - 4ót - 3111 -o 3528 324 3~78 4!56 3618 72 3678 336 3118 144· ss2.a. 33~. .3179 .U6 .. --JÜ-9- ts-o-•-- _,_._,._ -.0 -- !'19- 480· 3!53Ö 624 3,80 0 3$30 128 3610 240 3'3Ö 336

-'":u 3& 4_,_ -~~~1------ . 6oo ... .. 36it; . o a6es. . 96.0 ---- -· ,_,,1 -- - --.'-'I 3S3a 12~ 1112 1ao 36~2· 120 3612 3&• :s1:s2 1tt·

_________ .. JI33 ..... _S_~_a __________ ll .. l3. _____________ .. 0 .. U.$3-~-- 2!8-- - !683 ---240- - - -- S'33 li-6 3534 672: 3114 48 36~4· 188 3614 240 3134 984

. - ... 3.,_.3.1.... - ____ Q___ .3..11~~-. . _5_26.. 36.~!! 240 3615 480 3,35 9 3536 1,2. 3,86 384 3636 104 3616 !18 3136 !!2"

_ .. 3.13.1... --'~ L._ _ __ J.I.ll _____ ~ ... !1..8 ........ -----3-li-'1: _____ ~- .. 2.6.4~ .......... 3.6l'1-- ... -- --0-- -·-- --3'1-S-'-~ -- - . ~- '1-6-8 - --3938 336 1 3t88 102 3618 120 3688 216· 3138 281

.............. _.35_3_i_ ___ .... -~---'_!_2_;_ _________ 3~J_9____ ~JfB ... 3t:Î-f'..... ------8----------3-6--19~------- -----9-6tl ------3,J9.: !64,, 3546 aes. ~~~o 6?2 S6io 102 36to ''6 314i o 35~41... 4."6 !li~1· 0 !64!. 012 · S60t ····· 3l·l 3141 '61 3542 ,,, !f92 144 36.2 336 3602 240 3742 288 3!43 .. - ... Q.. - -3-.f9-3-- - 489 - S6-Á3 216 - 3603 -3:56 · 3743 o 3144 216, 3194 612 ,,.4 0 3604 432 3144 216 !545 864· 3195 112 !64' 648 -Ut! ·- ·it· 3145 2fJ8 3!46 6ÖO 3996 336 S646 384 3606 0 3146 91!: 35.4'1 216 319.7 314~ !6~1 G 36.'1 240 314'1 188 3548 o 3198 aas 36•8 48 3608 116 3748 24ó

_ :!!549. ... 110 '''9 . o 3649 ••o· 36!9 1"6· 3749 612' 3!56 336 ,,00 1,0 36!0 060 3100 168 37!fi 600

Page 18: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

__ ... ~-~-.M!.!.!J}_: __ ~-- ___________ N._ ________ ~ ___ liU.M._ __________________ Jol _________ .. liUM. --- -------- --~ J>(_________ J'11 . .U~-- - - . .. -W- N-UM:

___ .J.,,1 _____________ o _________ s8_o 1~----- --~•-o _______ :s_au: ________ ,at ____ !tü _______ .u ____ _3;.u_____ _ __ --D------------------------------------------ :nsa - t9z· 1102 264 ~,,a- 216 19Ö2 11a 3912 o

-----------------·--·-·····-----------------~'151___ ----'"'"'-~--------~· 0 :5 __________ .3_6_.0__ .. JII!_ .J12.. _ ~9Ji3___ ..... .D_.... 3913.---- . 12-0----- ... 3'154 408 1104 0 38f4 488 3904 72 19'4 "''

----- ---- --~;:1- il~- ·· --~=~~- - i~la 1t1: ~·:- · ~~s:-- ~t:. ~::; ::: 1'151 . 408- . .. 3807. . ...... 0 ___ IJ.Î~ __ '1.68__ _ 39__ÖJ' ...... J.'-8· ____ .3911 28.8.-------------------------------------3 'se- · -·a 16:-·· -- --~~ óe- --- - - - ·-- -, 6 - 1 à" a 4 8 ~ :s 9 ö 8 2 4 o :n 18 2 6 4

3?59 0 1609 768 llll 148 _________ 3909 . EU.6 ___ !919 ó

~Iltl _____ ~~~:r_~~~~-~~-~~::rr-.~~~~-~~~ -;I~ - - :l:t --~:: ~:i~- ~~~- .:::1 ;:: 3?62 480 3112 384 1862 112 3912 218 3962 ?20

__________________________________________ 3'1_6!_ ________ .L4_4.- ____ ...1l1t~----------'i~.o .... ____ __3Jil __ _____ _a_____ _ __:s_tl.s~---- _ _ ____ .a.aa __________ .J.t-63- __ .. ---1••·--3'164 ''2 3814 !152 316• 288 1914 '168 1964 0

________________________________________ ll.6.! __________ 4.B.D ______ _!_UL________Jl ____________ UAL----------•IA8--- -----3.9!5 ~-- ---------!!4'----~-- ~-~~~- - ---~-'--'-'•· --------3766 432 !ll6 360 3166 112 3916 191' 3966 120

~ . .31&.7 -~ ·-~- !15.1 __ p___ .w.. . p _,...,_ •• 33.6. - .!9t7 -~b4· _,_,,, . -9 3768 1~•· 3818 710 !868 0 !Oi8 1!8 3968 96 .

-~------------------------·------~--------~--------------- ~--1'16.9. ___________ !.U _________ !119 ---------3.14 ____ -------3.f-4t ~--- -1U-4--~-----U-î9-- -~- ~0-- -3969 -·-- ttl -- --37?5 864- 3120 96 3870 210 3tto 168 3t75 !36

------- --· 37li .60_,_ _____ ---1&21~---------- -~11-fS.. - -- -- -38lt.· 0 -- !021'--- ---· -431~~ 3971·- -- ·- · -594 !7?1 0 3122 432 3872 i!l 3912 240 3072 144

.1723 718. .3113 -0 31~3 J36- 3913- !JI-1 39'13 140 3774 ''6 3124 ó s•t• 114 3914 3•ó 3014 1176

...... 371:5 _______ .Q___ __ll.t! 1tlQ8_ .147! 576 -- :Ut-5-. - - .50-A· 3.91!S· ll 1776 12: 3aa6 3&4 3&t• 188 3916 iaoa 3976 101

_3?'77 .... - . 33l . .312.7. !2.8. - ,,_,,, 160. :ltl'l . - 0 3977 ,,. !7'18 aas 3128 112 3$;8 864 !918 110 197B •ao :t17t .1. .. 44· !11.9 - 48.()- st;.l -0 !O;Q..._ ----- Î-0-1-4 39'19 480-!?SÖ 384 3830 !SIS 3880 ~·· 1930 480 3910 191

~----------- --~------3.?8!.. ~.336, _________ 3J~11---- ---0-- 3!8!· -164--------3931-- 264·---- 101-t 816· 3'182 5!1 3832 192 !812 288 3932 0 3982 336

------·--·--3~'Z-83 ______________ Q _______ __3l.3.3 ______ --~-!.28... ---!éf3.. tJ~:.... -~--3933~-- 7-10---- 3983 0 3784 1,2: 3834 1112 3884 360 3934 384· 3984 144

---~--UM. _______ 5.1_i~----------.lU.!: ..... ~ . .!!.8 _______ 1-ÜJ~- 176- !QJ 0- 3-0I!J· 281, 3786 8~6· 38~6 0 38.6 432 3936 144 3986 110t

----------·---·------~----------·------~-------~-------..Jtfi2_~--------------·-11-l'------------~_l;t7 _____ ~~---'.36.. _____________ 1~U~--~---:G.~---------3.0.!-'-------~--~-2-A.8, ________ ~J9U-------------3.60---3?88 1!,0 J8~a :s36 1e,a to• 3tse ''' 1988 16&

----------- - - ----·----------~--- -~~=§ ---~g}-------ft::------------~-- ---'~:::· . -,_:~!-· --·---~:!:--·-· ... -="-- ----:::~---- ti~:: 3'19i Q :U41 .. .5.7.6 31tt.· 1'14- 1941 'IÓ- -:J90t· - Ó 3793 144 3142 480 ~892 l44 3942 384 3902 3ll:

- 3193 4.3..2: .. .3&43. -240 3103- 418 .. -3943 0 3913.. ,,, 3794 960. !144 198 38~4 5,6 3944 118 3094 360

.. 3.79.5 .. !84- !.lA! __ --- - 672 SIOJ 0 -J948 480- 3995 - - 6!11· 3,96 1~4· 3846 816 3806 432 !946 6ÖO 3906 b

. ... !1.2j_ u 6 --U-4'- o. 389'1 '!a 3941 468· !90'1 28& 1798 36ó !a•8 3!6 3808 !12 3948 102 3908 480

.. --31'9Jl --- 0 -- - ----3!49- '1lO 3809-'-- .. ,1(, - 3949 .. 384· 3999 G -3100 504 !8!50 480 3tÖO 0 39!10 6'12 4ÖóÖ 144-

Page 19: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

.N... ..N.UM___ -----------· JL . MUM. N. ------N.U.M - ... _.,__ NUM

--------------~-------~--------------- -:~i}--------:=:-. :::t--------- -- ~::. ··- :f~r :!: :i:i: 1••- ::~~-- ---- -:::- -_____________ 4.ölil ______ u.2 .411:! .31.4 _ •t-O-l- --O -----'-Î-13--- -34-ó---- --•a-93------- 1&4·

4004 48ó· •• ,. 416 •i§• 384 4114 i111 4204 120 .. 4ii .. Ö! 480 4jl!. 0 ''-!' 1-16 4t-H 238- 4205· 606·

... :J~; ,,r :J:;_ __I:g _:tit _:::_ -!i:~- o8t !i~~ 6':· 4oó8 192· 4658 648 4i§e t6 4118 314 •eó~ o

_ _ __ .4hij9 4~.2.: -461.9 - 720 .At,t)O 1t0 4 -4t-l-9 · .0- - - - 4209- -- 6!11' 4Q~ö s:4: •i•o . o •tio_ J&4 4160 •• 421ö 336

.... 4.Qtt . .. 4 .. o. .. 4t61. .. - tto -'-tlS. . -• 4161 ''' 4211 J!Jt· 4Q12 9§; 4f62 336 4!12 t92 4162 24ó 4212 408

----~--------------------------· ------ ----- ·---~iil--------·~:~~----------tt-1~-------------ilg~---------~tt!--- ---::t----··· ... --:i:~~ ------~;:" -------:~i! ,:ffl···· • ö 1 !I __________ .i _________ !l6.t_ ___ -------~l~t _________ 4iil~ ___________ l.ta __ ____ --•1-U-~ - ----- ---4-lf-t:__________ •a s.s -- --- ---- .. . ... -o --------------------- ----------------- ----,oi!;--- 1~• 4166 ,.,, •iït 384 4166 112: 4216 192'

- ~g.i?... ,_, __ '-_'- . -~.i.t1 .. l.lt! -·- -- ·~!1 •· 288... .4-1,6! . ········-0-·--- 411'- 414-4018 336 4868 288 4118 612 4168 144· 42i8 288

_______ .4_nit ____________ u.i~---- ____ uu~------~----•-•o _ ---•i. 9.... ___ .__ ----4tto - lo.0-8:- ... •219 nö, •ö2o 192· 41'7o 864 4lio l44 41 '1o ''' •eaö o tl\021 .. 7i14.- ----•i-1-1 0- •iit:- 1186--- 4-1,1- ·-!14·· -- 49-lt -96Ó . 4Q22 648 4j72 216 •iit t36 41'12 192' 4222 288 -'023 .0. .,,3 672: •-tt3: !92 ·~'13 314· . 4213 6 4Ö 24 240 · 41'14 480 41 t4· 0 4114 !S!I 4214 96 402!1 .960- 4f15 168 ·~·,, !176- 41'15- : ~- -•219· - . 3tÖ 4Q26 ''' 4Q76 312 4tf6 384 4116 216 4216 1248-4027 ~.16.. . -412-7- !U'6 -4111 0 4117 410 4!11 14ft' •ö2a o 487& 336 4it8 96 4118 ''t 4228 102: 4Q2.9 480 -4f7t -o- 4-ttf ','JI · 41'19 - 3-1-4· 4Ht· 936 4Q3Q 4SQ 4tlo . o 4t~o ''2 •110 101· 4t35 120 403! - - -OL- 41!1: ____ -- 5'16: · 4tlt 614- - 4181 7!-G ·-·····-· -•2-11 - '0 4Q32 D 4182 624 4f32' !44 4182 672· 4232 300

__ 4Q.3,3. __________ a!L_ -- ·-- '-8J.3,_ -24-0 -· ---41--$3---· tS6 -41 tJ- - --&· ···· ··-- · --41!3' - -480 ---

----~~------!H - nn --:!i :m ::: :m :H :m m ---tr.-.3i-- --.--. ----f_ió-------_ ---~:•_,:t--~-----i~~----------~~-::--·----·-···----·--··~. ::----------·:it:--------------- 3"~"--------!:::--------- --- -·~::-·-··--

4 0 3 9 ______________ Q __________ 4II.i ___________ 4_8.Q ____________ 4Î.3J __ ---- ...... A IA.. --------'-1ll--------···· .. :4.31- . 4.23-9 --· --··· --·· 0- -------------------- iiil-- 336, 4jto 288 •i~ó 240 4100 8t6 424Q ''

~0~1- .1.~ .. 0 41.91 .712... 4141' 336 41.01. 0 424! 1206 4ij42 432' 4192 o 4i~t 624 •102 12 4242 4&d

. 40 4.3 .. _:S$_4_ 41.13. - -- 31.2 .41,43 0--- 4S,03~- ,,_, - --·· .4243- . 216 -4044 288· 4j~4 1112 4144 06 4i04 480 4244 311 4Ö4!L . - 480 .Aj95 0 4-t.l 5·16- ·- 4-iO! 192-- 4245 _,,,, 4Ö46 912~ 4j96 6 41., 6'2 4196 518 4246 432· .tii-4' ~, 4i91 314 •i11 2&8 •1 ''- "• •a•' p

...... -- .. !i:: -a!:. :t:t ;:~ !t!: 1::~ !i:: !ot;.. :::: - ::: 45~5 '~& 4~oo 4Bo 4i•o •oo 4t6o 336 4215 &64

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- .... N .. . NU.M

4291 384 •so1 1DI6 •t11• e ••ö1. ______ ,_to __ ------~411.' ________ .696 ------------------·-------------------42~f ---------o------·4soY·----------4ao- ··--··si2-·-----·· 48···-·--- ----44oi____ 384· 4412· 1ta

42!33 !!JO 4- 4803 0 4383 1~:8_. 4_45.3_ ___ _ ~IQ _4~1:L _ 288 --------·------·-·----·-·---····----------------425'4 -- --9-12: ______ 480-.. -- 264- .,,. 5'16 4404 336 4414 9111

~~~-~ . J . . _ ~~Q-~ . 3$4 ... 4•j!l 4f..D 44ÖS 218 .. .. 445!5.. 0 . -----.i~ 6 2" 0 41 0 6 4 3 2 4 SI 6 3 t 0 4 4 Ö 6 t 1 '1 6 4 4 I 6 2 6 4

-------~~~'!.- 4:n: __ 4ïo~-- 43_2 ~tt1 .a6à__ 4417 a _.._.,,.____ 1:ao-42!J8 288 4108 288 4388 104 4408 240 4418 576 --- •a.'' S4o_ _ ___ •-~u _ ....... 432 _ A!!lt·_ .o __ . 44öt .,,, .. •••9 - ..... - z•• 4260 192: 4!10 612 4:160 244 4410 840 4460 240

_ 4261 4Ji!.:__ _ _ ___ 4U1- _ --4- . 4U! 1~oa . ••is.- . .a a a. -.4«~- . ''t· 4262 &4~ •!12 216 4$&a· 432 44~2 o 4462 •eó

_____ 42.63___ _ _ _o_ _____ ___j_u_a _______ ,u ________ J..tl.&~-----------.IU-------••s.s. - ---·--- .!t&------- . - «43:----- --- -- ----Q--- ---4264 2JU 4!14 48ó 4364 •oe 44SA 384 4464 o

______________________________________________ 421!5 ___________ ,_6 4-_____ uu ________ !_3_6. ______ --'W..:. ___________ _,__ao. ___________ ut.s-------------···---.0.'-- --------'4-6-J-------- --4.ao - -4266 960 4316 0 4$66 614 44!6 96 4466 768

__________ 4.2 .. 61 .... ---· ___ _1!2_;_ _________ tl1L .. --- --- .tJ.A .. - .. --- .4.S.U - ___ fi 44!1 -218- - 4441 111--4268 28!· 4S18 384 4!68 96 44i8 '88 4468 168

·------··-----------~~~~-~----------·J~ !;~~~-- : :t~--~---~~--"191-- ------····------~:tt:--~---------- !-~}-- ··--···--·---!:~~- ·-- --1~::- .. --------·····-~:;~- ······· . ----i;-::.<-.......... . ---~2ïi· __ o_, _________ ~_!.i.1 _________ :t..8_4 ___________ f:ltt ... __ __.a a 4411...... loeo .447t· o

-- --· .... 42?2 4! 4!22 912 43;2 SlO 4412 384 4412 431 ~ . .2'1:! aJU . 4!23 19.2 4Si3 . 8.40 4.413.... .o .. 4.42.3 &64-4274 toös 4824 192 4jt4 •66 4414 ''' 4474 :sta

.... -------·- _4215.. . ........ &.ö_n __ . ____ tJ_u ·----11'16 -•l7.J a __ 4j.t! .............. 3.J_. .......... 4415 --6o.o. 4276 360 48&6 768 4376 !12 4416 3il· 4476 0

_ -~t17 _________ ,f."l ___ 4.827__ _ .0.. A!.Ï7 f2.8 . 44.17 _ .. :U4· 44?7 · 3tt: 4278 384· 4828 264 4378 432 4418 ,.. 4478 576 4279 . 0 4.12.9 960 flit' 384 4419. 72t) "''' 0 4285 432: 4~30 432 43~~ ti 4430 i104· 448Ö 48 4.281 !21 4$31 8:1.6 4lÎU.- .JlA 4431 0 -A48i ?63· - -4282 jf{ --- - -4.32 112 -- -----.!11~-- -- --- '168 - -.431-- - - ---- ,-. ---·- 4482· 384

_____ 4283 ..... _ .. _______ ,_n.4.:~------'ll3c ______ ---3-:U ... -------- '-3.-l:S.:. _____ - ---0-- ... ----A433 --- --·-· --6'Jt· - 4-483-- -i1j -· 4284 0 4334 91. 4j.. !44 4434 576 4484 518

........ 4ll! __ __ _ ____ ut~ __________ 4.U1L ____ . _ .n .. __ 4J,JL 1-tsa.. . 4435· . 2•0. - 4415· '''. 4286 960· 4836 0 4316 576 4436· 60Ó 4486 60Ó

-~:~i---------'--~~~--- ::~!-----~---~~- ···--·--- :~;:-----~----::-:----·'-·---~~~:-------------::t------~::::•~---···-----1;:"··-----···· ...... 4.2.8.9 .. - .J:Jj_4_,_ _________ .AJ39 .. ------------.4.0!------ ----4-!A-9 ----·------1-'6---- ------ - 4439~------······- · - 0---- - -- 44JQc.. · ··4!f,--

429Q 516- 4840 384 4300: 288 4440 101' 4490 1086-4291 2!8 4l4i 614 4-30!. & 44.!. ,,, 4491' 360, 4292 288- 4342 240 4302 160 4442 4!6 4402' 120 .4293 .64!, - 4f43. . . 0 -4!03: - - 218- 444~- ~36- .... 4403 416· 4294 432: 4844 336 4304 1808 4444 D 4404 864· 4.2.9.! . 0 -- 4!'5 !16 ,,,, 480 444!5 768 -- .. 440!J- 0 4296 384· •••6 1200 4396 l44 4446 ioas 44t6 a•o

.. U.i!. . 432: 4áj7 . :114 4iJ' 648, 44.1 G 4401 431' 4298 528; 4~48 0 4308 136 4448 168 4498 336

. 42.92_ -- - -- .'l_3_2: _______ 4l49 .!04 •i••-- - 0 .4449 614 4409- 116 43ÖÖ 168 4550 720 44ÖO SlO 4410 612 4!00 431'

----~---··----------·-----·-------- ---------- ·----~---··----------·- --·-·--- ----- ---·-··--·--·····-·--······--- . ·--- -------·- - ····-----··---·---- ...... ·-- ---------------

Page 21: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

JIUM. .!\! ...

•so1 ___ __ _ _ 6R4 ----------~'1 a _ _ ~-~Ö1 .11.6 46!1· 4öa. 4.7öi~- .431: --------------~------~---------- ------ ---4so2 ___ ---~----"64&- ,,,2 144 46oa _ 384 4612 168 • , o2· 384

•sö:s o 4l!!i:s ~lo_ 416.53: tAs .46J:J_ _ ______ ,,_o__ •?-O.J:__ .d --------4504 264· ii54 1008 46g4 0 4614 !18 4'104 168

4!50! 168 4f~!L _ 2'-S _460.5 761. 4AIJ 0 4'05 480 . 4"5ö6 912: 4116 384 46Ö6 6'12 4616 06 4'106 81~· 4'21 Jti _-4;11 .4~2. ,,a,_ __ __ Q_ 46.17- _ aas -4'.o' -6oo 4!08 D 41!58 240 46Ö8 36 4618 '168 4708 144·

________ ,tlQ2_ 7_6l ____ --~~~~--- ______ a _ __ _ _ .4..UJ __ ~_aa_ •ug U6- -•-'-O t u.o•L 4515 28! .,,0 102 46{0 614 4660 240 4'11ö 768 4511.. D 49-'1 .... 4 .. 32.~ ---------- .AÛ1~ 672 4661 itö4 4~1i 0 4512 96 ,,,2 864 46Ï2 i02 4662 ''' 4'112' 191·

________ 41.1.3.. _____ ~ ___ ...1.8 a ________ .!.H3.: ____________ 4.16.... _____ --•us.~-- 48.0- ... ___ -44-6-3-- . --- -- ---0- ---- ---4'1!3-:.. __ --- --, »6·--''14 :t~4- '''' 288 ''Î' 4&o 4664- 4Ja· ''t' 6oo __ ---------------~--------- _______________________________________ J.!-1Jl. ____________ JJ.42___ _ nu _ _ ____ _ __ ,_2.,____ __ ___ -4-6-t:.J-- .. - . -------:"-e- -- • 6 6' ---- --- __ , ~~~ ---- ------47!'"--~-- -----!1.1-6- --451• 1~2 4966 432 ''Ï~ 672 4666 512 4,,. 360

-4.5..11 - . --- .. 14.!-'-.. . u.A1 < < -- ___ .(i_ - - --- -- _ _._,,_,._ < ,... • --4 ,,_,_ . '"' 4'1 1 '' 288' 4!18 s•o 4t68 216 ~~~8 1i2 4668 a 4?18 7&1 1

----------------------------------------------------------------·-----------4.!jJJL _______________ ~_Q__'__- ----41..69. ________ .li.0---------------------4-l!·----- --------U~------ --46-6-9C----- ····------------D•-------------4-119--'------- ----------&--4520 528 4170 336 46iD t92 4670 816· 4'110 0

---- 4!2î- - 8&.4,_ ------ •--11--l--- -672 -46-ft~ -ll-2 -- - -- .,,1 ---------------- --o~- -------- 4111· tos6, 4!22 288. 4f72 0 46à2 614 46'-2 48 4'-22' 48Ó 4!523 S.Ö4' 4t'J 3!6 4--6-Î:J 8 ~6?! 614· .. .,.3 116· 4524 192 41?4 1112 4614 t02 46'-4 960 4,24 1!1•

::i: :~î: - -- ::~: 14~ ~i: 1t:: :!;:- - :~ :;:r -- ::t 4527. J ---- ,,,, 768 . -•• __.,_ --1-4-ê- 461'1 -480.. -- .,,;_ 0 4!28 12: 4,?8 480 46i8 136 4618 216 4'-18 240 4!2.t . ~6o ''·'• 24(). •6-it -JU- 46,,- o 4?1-t· 4&o 4535 480 4f8o 218 46~Ó 432 4610 240 4?36 960 453!. 288---- 4f8-1-- --1040-·-- - ----46--!1' -- -t . 4611 _,,, - 4-1Jt· 480 4!32 336 4'82 336 4632 218 4612 ,,,. 4132 o

---4!33--------- ------21U.~---- --··- -41.8-3----------- .. .g -------463-3 -- ---- --- :1--1--4-----· -46-13-- ···--- --514· ·· - - 4·,13 - ·--H6 4!34 648 ,,.. 384 46!4 864 4684 168 4?34 '-10

.. -- 4!.3.~------------· ---------0.'----·- 4ft5--- tl--8- '". 432 4685· 118• .,,,, 0 4536 432; 4186 1032 •• ,. 0 4616 6'-1' 4,36· 110

... 45..31----- ... aas __ . _______________ ~._,, _________________________ e••-------·---4•'-----------t36----"--- --4-68--1 -- ---ó---------·-- _____ .,.,s7·------------·--·-------4St·--- --· 4538 696 4188 !44 4,~8 432 4688 216 4,38 384-

-------------------·------------ ---•sa,________ uQ-, ·--uao-------too8-------------••s•-- --- - -46-!9--'--- --ïs.1s·----- .,_.,,_ 6!4--4S46 ó 4!9o 768 464o a•o 46tó 3!4· 4'140 19~· 4!14{.. - lt96.•. - 4lll G·- - 4fl.!: 168 46f1 504 4'?iU· 480 4542 5!8 4192 0 4642' 288 4602 218 4,42· 744 4!543 ---~'------ Al93 33.6 . --46~3- 312 4603 ·414" 4,43 0 4544 G 4194 528 4644 "'' 4604 ioao '''4 281 4!14!!1 ,eo _ •••s 516 ••••· 314- '''' e ,.,,--5: 96o 4546 480 4i96 102 4646 1056 4606 36Ó 4'146 ,, •. 4!!47 4Ö!~ .,,, .• ,, .. ,.,. 0 4-691 ,_,,, ., • ., 191' 4548 2-0 4198 936 4648 144 4608 408 4,48 216 4!54--0.. .606 -4190 0- 46-49 - 1016 . - 4609 118 ·4'149 960 45~5 1ó08 4100 240 46~0 '20 4?66 0 ''IÖ 288

Page 22: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

.... N NUM ______ D~- ______ 4LQ_1"-------· -~-l2 ___ __.fll~------~--'-48_ _ ____ 4gjj, ____ ... ____ tJ.6- _____ f_tJi_____ -- ---·- 0-- --- --.

4?51 192 4802 ?8Ó 48f2 SlO 4902 384 4912' 504 4?!;,L -~:t~ 410._3_ 336 41!.3. 'l.68 _ _4fÖ3 j_ 4.t!.3 _____ 3.1.4

----·---~--------~--------~--- ------ 4754 912: ---- 4104 ---it2' ...... 624 49Ö4 !S04· 4954 600

__ 421,__ a_e.~: ~-J.oJ· .'l.t2 4&f!: ____ o 4tös itlt~ •-•"s 611: 4?56 2~0 4106 96Q ... ,, 480 4006 431' 4916 0

--~1!'l_ 6&4· 4107 --- - 0 .... ,,._ -- 4_:12 4..9_01_ -- '288-_ -'-''' --- ·----- - 3!1 .. ,se - ----,-,~:--- •aas·------ 218 •••• 48o 4958 ••- ,.,,8 ·~• 4?59 _Q_ _____ 4lO_f____ _ ________ et~-- _48ft . 120_ __ __4909 360-.. 4919 o

----- -- 4'f6l 480 48S.O 384 48-Ó o 4910 i1D4- 4960 96 ___ ~.l.li 7~t- . 4111 ___ _ __ '!EL _ 4.A.A1: tt6 . 49i1 .. JL - 49.61- ÏO!J6·

4'162 4ó8 41S.2 144 4862 480 49t2 '12: 4962 !?6 -- ------·--- ------~?:.lll__ -- 4 8_Q_ _______ ~~-·ll __________ 36_0 _________ ~Jil_-_______ ---·--·-~ _____ 4t.i3_ . 8 .. 64~------ __ -'.963'--- 281.- ..

4'164 0 4.S.4 1296 48~4 24 49{4 i111; 4964 384 -~----~-----------·----------~-4 '1 6_~ _________ _4 3 _2: ------~--4.1U_~--------- ___o ___ .. ____ AlH _____________ 6.!2___ _ ___ 4_9i! __________ u_Q ________ 4~.4J:__~- ---!16-:--

4?66 960 ••s.• 96 4è66 612 4916 416 4966 671 --------'-Z..6i ____ ·-····------fL _________ ffi.'L ___________ 6.2.2-_____ Ü~t:______ --1-1-2-- .. 491,7 -B'16c..... ~--40.6o;· --G• ..

47&8 12: •Ate 314 4868 314 ••te 264· 4t&e 384· ·-----·----~-------~---~~---------~--------------~i'.61 ___________ s,_D__6_j_i__ _____ ~üll_~----···--·----•-32' _________ _tjAJ_,, ________ ._6.Q_f1 ____ 4fi9... . ... fl----·-····"'·-.4-96.9---···-· .• , •. ___ , __ _

4775 toös- •aao 480 •s;ö 1!8 ••eo 218 4976 671· ....... ..4.l1i . ____ 2S_9, _________ 4ll1.:.-------- 672 ... ------48-lt.---- --0---~- 491:1. . . _,,,, 40-11-· .. 384·

477t 432· 4822 416 ••fa &8$ •022 121 4072 96 41.Z3 4SQ 412! -·-~·-· 4-1"7-J: 148 .4913-. 364 491!: 79J: 4?14 480 4124 !04 4$;4 188 4924 0 •• ,. 111

__ 471! ____ D~ -----•••s- __ U6 _______________ •. $-l'- __ ~--- 5'16 .... ' '" . a•~- tt''' o-4776 336· 4126 t•4o 4876 t44 4916 7tó 4971 o

. _47.1.1.. .3J4.: -4121 -218 4Û7- 1'--2 .49-2'1'- 0 .... ,,,, 410 4~je 504· 4128 ö •tf8 864 4928 •• 4918 211· 41.79 64.9-. 41-U- - 12--48 4l~.f .tJ 49-t-9 -· - -612' · 4919 ?IÓ · 4785 192: 4130 384 4è80 t92 4930 314 4985 384·

__ ___ __ _ ____ 47 e.L _ . .J3.i4_4_ _ __ 4A31 4lt1 . .. - -- -~224---- ·--·· 49ft. 8-40- --4911 . :JI4· 4782 33&· 4832 144 4112' 218 4932 288 491;· 1016

·---~-·z~.1________ __ _.á:.. _________ !a.l_3 ____ .. _5_'16.... .. ___ 4a._s~~-------•·:s2.. ---••:~3 .3l2--------- 49ts- -ó 4784 192: 48!4 432 4814 384 4934 i272· 4914 191·

..... ____ !.z_e_;_ _ ________ ,rt.~-----·---4-B.I!~.-- lll .... 488.!5. !128 49.35 ó 4911 ,, •. 4786 480 4836 480 4886 1808 4936 288 4986 791'

--------------~-------~------------·- _____ -~----~1_8"!_ _ ----~ U_Q_.__ __ ~4U.1 ____ ~-~---4ll--------------A.lU.: ___________________ -8---------- 4tP- -flt-:----~-----4-U1--:------------116~---- ····· --- ------- ...... 4?88 240 4138 480 48,8 ,44 4938 4S2 4018 0

·-----------· _________ .4"!8CL --··· ··-·--- .l.6IL ______ 4tl9: ... ____ ------0---·----·--- -41-1-#1----- -1-011-6---------- 4-919-------- 3-84-- · ~~---4N-9-c______ lGQ8c-- --·-· · · 4190 ,,,, 4840 264 4800 384 4940 288 4996 128 A79i. _ .a.~ •••s: ''' 48tl 489 4941 !tt-4· 4991 e 4?9~ 1~4 4t42 600 4892 0 4942 288 4992 48 4'19:!. !t6 .. 4·.... 4-&43 112 .. •a-Os: - tao .4943: ........ o 499:! 314 4194 768 4144 336 4804 480 4944 144 4994 11!1' _4Z9!L _ 18.! __ 4-.1.41_ . 384 -48·8-- . 0 - 4945 41-0 499-J - 384-4796 0 4ê46 '28 4896 240 4946 960 4996 384 .4."! .. 9.1 432: 4A.4'1 o 4J-01 4!3 4947 211. 499'1 1tö 4?98 ,,, 4148 0 4891 864 4948 168- 4908 431: 4199__ _ _____ _n ____ 484.9 . !.1.6 48-0l--- ---114 4949 Î1'-6' 49t9 D 48ÖÖ ~6 41!10 1200 49ÖO 270 49!0 840 !SOOÖ 444

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__ .Ji_

Page 24: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

__ _,'41.J.~·'··· -~-~N ~-~-----~--N.UJL _______ ~---~--N.- NUM .. Jt ..... N NUM

!IJ25i 33! !l'Ll~-------~-~~0___ !Sit· 0 !J4fi1. .... 4l2 . . .... '4"-L .. 6.-'l.l-~~----· --------------------- ··----------~--~~-~--~--~2si·--- aas: ssoa 336 -i:ii2 2it,_---------i4o2_______ ,,.. '•sa· 1•• !IJ2!1J3 6l12: !!J):J 0 5:i.i3 ___ 1.7_6_ __ ~----~--UiJ.: _____ 314 __ !Af3__ 6-'l-'--

---------------------~-----~--~-----------·----~~5~~ ,~,-,~- ·-----,-tó4 ___________ 4äo-- '~i• 14!.6 !I4Ö4 ó !J414 '''

--- --- .!.2.~!__ . _____ o_·.. -- ... W.I. --- 48.0 ·-- IÎP' .. 411) .... -- - !4Ö! 16! 1.4.1-5'- ~· !J256 576· 1806- 864 !J~86 t02 !1406 410 !J416 336

--------~-------·-------------~--- ______ _ll~!~- __ ;J~ __ i' _______ fJ!LZ __ ~-----·· 2$8- ~---· IJ!f: _____ --~---:1.21i ______ .. 14Ä1..... 0-------- ---'•1-'1..:.______ -3S.---!J258 '"2; 1108 0 ,,,. 480 !J4Ö8 uso !418 4111-

.. ,_2!.to _____ al~ _____ !ltti__ 1.2.12 ___ !Jtt.~-------- O.---~ --'-4Ö9 i2o-é.-------'''-'------ _ &14-'2&5 1~4, 111o 1ao e:i6o e !4'0 sa8 !J46Ö t91

___ !..261 .... 9_,_4.- f_l1.1..... 0. Q-.61.. 22.0 !JJU.1 81' - ----- !.44-1 - ltl: !J262 6!4· !112 12 !1!62 288 !l4t2 192 !J462 I!JI:

. -~~JJUJ. ----~~-----~~.0-----~---~!.813" -·--·· -·····---~1!6. --------Il6-"------- - 31.4 --- . !41-3 ··- ~- ---l!.t.-:------$4'3 -- ·- ~ Ó !IJ264 ase !114 384 s$~4 104 !J4i4 i361i 1464 216

____________ ___ __ __ _ _______________ _!2__6_!_ ____ ~~-!_6_4.:.~. ___ ._111.5 .... ______ ---4l.2._. ____ ~QU.:._____ --- . 41# .. -- J4l!L~ - - .... ··----Ó-- ______ ,...,_'- .. -- · 101-6· !5266 624, 1816 432 1366 1110 !416 264 1466- 060

_ .12.A1. 3.36... 11.11 .. --~---- . .!U'- 0 14t'1 864, !1467 191' 1268 240: 1118 744 ,,,. 240 !418 864· 1468 ó

---~----~!_2_6,! _6,2; ...... 111.2 ___________ ~ _ _rj_______ !*t'' . 11!6 .. -- .lii-9 ------3!11 1460 . 671-~-127Q 672· 1!20 218 '$7G 672 1410 218 1476 431 .!927L 0·.~ ____ ltl1:. ___ ~---11l!l6__ _ ---~~~1 _ .. 2.88 -- .14.t1 .168 !!4"11· -0 1272 120 1122 384 •• ,. 0 !J412 336 14"/t 216

.1273 480 =.. 112.3. 360 IS;!.. !1!6 . !.4t3. . 0 S4':J; Ué !J274 1080 ,,.. 218 '*'' ''2 !414 144 ,,,, 1!44·

_ __ _____________________ --~- !!7.5 ___ U~---. --~1-!ll:_________ .612. .... .!St!L . . --Ó---------.5419 ,,.. 14'! - - -41-ê-- -!5276 0 1126 624 ,,;, 48 !416 624· ,,.,, 2tt·

.... !2'Z7 ________ 6.L4 _________ !1at ---·-· .o .. qi1. ··-~·----2•-t-~----- ut'- 40-8 -· .. -· _,,,, .. ----- .. as a 1278 aas. ,.18 110 ~tfs 768 1418 102· 1478' 384

=~~~ 9i·· ·:·11:- 1::: :t:: · ::: ::::- :i:· :::l !U!~ !2Bi __ ·-- ... _____ fLU~-----~-·····1-131: 410 .... !l~i-1"-- -·~-~----11.S;2 !1.4:11· 0- · --------I.Ui'-·· · "168 1282 ?68. 1832 t4o !l3'2 ••o !J432 288 1482 3t2·

.. ____ ,2_8_3 ________ ..lfl4-'--.------· __ fs33_ "1.~4. _______ .JU"----··----····· ... o. . ... 14:13- -- ------432~------- !l-413.- 408-1284 288. 1834 960 IS84· !36 !434 384 1484 288 !2.8.!.... . .!:6.4:. . -'U-!- 0-- . .. •••• . ' '' 1431 - 624· 1411 ·~· 1296 528 !816 384 ,,., 600 !J436 0 1486 1302:

----------~.287_ _ __ .a....... Jtr5'l ---··------a•~~---·-·----'"'.:___ .. tsa -- t4-J'1-~- -1-tt---·-----··--···· .,,.,.,. . o ~288 5Ö4- !~38 288 ,,.. l44 !438 960 1488 0

····-··--·-··--------------J'-280_. 48.Q. __ __)SJ_Q __________ ,!-1-6.---- -- '*''- .. '120-··~-~1439~-· 0 !J481}. 1008-5206 ssa: sa•o o s3oo 1oós 1440 •a •••o 120 S29i .864 !t•s.~ .. 6.48. .. !192.· ~ 14._1 it-6 1491 4-QS, 1292 224- 11-2 168 JJOR 96 1442 612· !492 216

__________ .. ___ ... 5293 a.ea. _ !.lU--- ----- - - o 13o1: .... a~-4- -~---'"' --- -. - as.• !5403 624-1294 toas: 5144 144 sso4 4&o '••• 11ö 1494 sta

-~------------·---------~---·------ --~- .!!. 23. _ _ __ .0' ____ . ----'4'3-------------~--.!.76 ISO S -~ US- - . . .. 1448--- -- -- . 9-36- -I 49J'------- - ... ----6- ----1296 12: li46 1248 !396 612 1446 5'l6 1496 3!6

.. JJ2.9..t...... . ... 8 .. 6.4.~.... . !.$.._,__ 312 ,,.,_ . -41-9 144~ 0 . , .. ,,.. 288, 1298 480 lt48· 288 !J3t8 36Q !5448 288 1408 ,~ .. _529_9 _______ 2!1~----------~f'9.~ ---···- .. !!8. ·--~1-S-OL----~~-----0- ..... -- . --5-44 0 .... 6'1' - .. J4-0-9·--·· ··12G .. 1300 504· 1110 360 1400 480 1410 840 ssoö 0

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. NUM ... NUM

55Ö1 SOS2: ·-·---·----~-fl1 _____________ 0 ______ '6.Ö1 ___ 220 _.56!1 .. 144·_. 510-1-- --· ---416.- ----------- -----------,-soä ____ ·--··-·--- ,,6, s~s2 12o ,,o2 336 16!12 :~•o ''Q2· 6oo '' ö :~ __ o- '.tu _________ 412 ___ .. lt.fis .. __ ,_uL . ...!.ll3:.. ____ .!a.o. _s,u:. . .. ---Ó------··-··--·-·····--------·-·-------------------------------ii04 12 0 · Si 5 4 7 I 0 ! 6 5 4 3 3 6 5 6 !14 i 3 4 4 · 5 7 Ö 4 3 8 4,

,5_05 4.~.0 lf.~5 5'16 !6Ö! 384 !JUS -0 5?05, 6'11· !5Ö6 !!I; 5~56 336 !6Ö6 1112· 56!16 336 l?ó~ 84Ó

____ 55Q 7______ _ J~l:_ _ _____ IJI!l _____ ---·- 4 .. 118_ .!65! . ··-····--·--- ----8--- -- ..!6-B-'1 ,,,. __ - --l?ó1--------- -1PI'

;:~: -----~:t ________ :t:;. 624 _itt~~------J.i:J. :::: ::!~--- i:J!~------- :;:, i5ir-- 864- '''o 2•o ''io ''' B66o a ''1ö , ••.

. . 5511 _ Q__ IÎ.6.1. 1514 - !6.it:.. t'-4 5661 itt6- - J?h.· ó 5512 240 5f62 5'16 96Ï2 336 5662 518 ''12 96

-----~-U3. ________ 612L_ __!.ll .310------ ---56-Ï~---------t-11- J4t63 -----ê-------41!3· 111· 5514 720 5t64- 0 !J6~4 384 5664 1t2 1714 960

____________ ---...!'-15-------------a•.o.-- '*'' ... _______ s_7_6 ___________ ,,!J'----··- O-·· --·-·· ,,_., -· ----- 4tf---- ---···-·-'11; -- · --'1-t o · 5516 384- '*'' 528 ,,,, 0 !666 t6ó 17t6 164-.5~.12.... 6.fto. tt67 ..... .o. __ ,._._,. ''' -'''' a•o ''1' •:s•-,s~e 21a stee o ''is 66~ 5668 1•1 ''te 121

----·-"·~---~--..!.5.19---- .... .Ql_ - ----'tAt-----~· ----!14-·---····-- -IJM~ - ---·-'-'2·-····- -U6-9C------ --l2-'IF "'19 __ ::i~t _,:t_ __iUf~-----~r: __________ illt _:::____ :!~~:__ ___ ai: :;:~. _ -~::-

''22 864 ,.,2 192 !6t2 184 5672 4ó8 1112· 486 5523 US· !l'l:t. 744 16~· S.- f -lf.'-3- · 410 5,13 -336 15524 35.2: ,.,. 720 ,._.. !528 ,,,. 360 ,,,. 0

. _ !52! 9.6JL _________ !H..5____ - 0- 1-itJ ....... -- -2-lf.----- --56-7!J 14ê- -·· · .. J,-tJ:. - 66G-----5526 &221· ,.,, 576 ,,., tao S6'l6 218 s~t• 11sa·

_.!52.-' __ . ___________ _n _________ m·L__ 12o !,,, ________ ---"-2---- --s''' .,.. "''~---- b 5528 456 ,.,. 408 ,... t) 5678 864 !9?,18 12•

-· ...... !1529. - ... %tU, .. .JJ'lt '1..24. . . 1-f.ff. ------ -----41-t--~ 16'l9 &- .J.,tt-c- 11~8 5536 486 s-teo aas si3o sa8 s6ao 16 s?s6 ''~

. !153.1------ ____ _s_n;____ -----···1· 504----- - 16$!-------~------0--- •····· -1611 . -'-168·-----~!·'1-St~ -- 4-80 5532 0 St82 118 5632 24 5682 516 5132 431·

.. J.53.3 ___________ :J.!.t: ______ .1113 ---o..~-------'-i.s-3---·- ..... tta ,,., ---264----- ---s.'ls-3·-- ... e~ó 5534 &Q56· St84 t68 56~4 1296 5684 648 5134 48Ó

--- . ~$.35 -·--- ... -- ~ !fl5. 1l'-2-.. ""i 336- ,,., ,.,. 8115 0. S536 110 ~~&6 864· sáse 432 5686 sta 5136 !Stt-

-- ·- !.5.37.... J364.· -!Si'l --- -l-'2---~----J6i1· ~~~ ,,,,_ -- - -- -o-- -- -··· ,_,,, ttJt·

_________ :;;:_ _:_:;; .. _:l:: ---------~4::: -=t~:" 4': _ -=~::....... :::. =~:: 1 ~::,

5545 516· 5t9o 314 1&48 384 S60o 864· !J714ö 96 !1541· &.64· lft-1 . 0 . J64l: 489 J-601 ,,,_ 1141' 11!8 5542 432: , •• 2 240 ··~2: 968 1612 0 5142 '20

. s5A:J G: '*''---- -344- 1643 384- - .s&o:s ,,. ''43 --o !5544 480. IS94 1214 1644· 06 5604 iOI6 5744 ó !5545 zes S.l-15 -- 218 16,,c_ 1544-----------$-605- -I) .. ,,45 ---- -- _,,.~

554& 96ó '''' o '''' tao 5606 144 '''6 '~o ;;:: ~~t -·--·-:·t;: ~=: ::~r ,.: ;:::· r.1· :::·:' :::. !1!4!- --------Ut~---------lt--•·•·-··--- o-- . --Jut·--- ----·6'lt- 5&!9 ''' ----------Jf-4-!----------- 666· ··· 5550 1008 5100 240 16~0 480 !J7DO 336 1710 144Ó

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NUM

0 -- ---~--'-"L--~----1.!~ .. 4~~~--·· --.f8fl .!Jll-4-- -----UÖ-1-:.----~--- ---''1' --- IJ911-· · 0---------·-!57~2 192: !Jaoa 432 IJ$!tt' o !IJtöa 336 !ltta· 41

-·------~-----~----------~------·---·-----'1.5..!. _l2.Q. ___ ~-- !80 3 ···---2-~CL----- Slf3~--·- .................... .4!2---~----Ui3-- --0.. .~J.tt.J:- · -·-·· -Jaa--!5154 4Bo: ••o• :112 '*'~ ''' !954 218 !984- iboe

..... .. .. ~f,~ . a!~- ···:·:t:~--- ~:: . ::;:· . t44 ::t: i::t ::::· ::~ _______ .!575"!__ __ -··- ~~4: ______ ll!L7 ______ .. _ ......... __ 0 .... __ IJ_IJ!:_ _ _____ '-32_ ..... _ !Jt.Ö7 . .31A, -- _____ !9-fi~- ~-------!l-61- --- -

~·~·---~-·~--··------------------ !1758 2B8· '808 104· '188 816 '908· 1PI: !918 101:

5?59 ·a .. sao9. 480 __ ~~--'~iit~------------'••-- !Jtät. Iaaa .. ~------'-''.t_______ d - j'76l - --·-Tao·---- ---!-ifö- -- - S.2i8 saaö st2 !Jt!o 384· '''O 4st· '''1 ''a: ''~iL zle nu: _ 7.t.a ''11 .a ....... !.961: 120 --,.,&i &'fz: Jll-~--... 168 !J8t.t 72P !1912 3t.O 1962· 384 IJ763 28!:_ . . '''-l: ~1~-------'··~:- ____ Q_ _ .IJÏ3 ... 8l'-'. _____ J.f.63c J'16 ---------------- -- ··-----------if64 ______ Iii --·-··-lli~ 96o !58A4· ' '' 1914 701· 5964- aas

-·---·---- ___ .!.?~6..5~-------~1 ~;_ __IU5_ .. _ _o_ _________ !jJ.!~-- 7.6J -- !Jt!S --· - -----ll-4c.. ________ _J_UJ- 4:11' !1?66 744· !816 710 lè~6 118 19{6 ~ 5966 ~Ó08

.~Jit6'l. J)_ !A17._ ---- .. -UA-.--- !.4•-'·· J-04 !!1917 -'!14- '''' ~ ~?68 285· IILB 312 IJ$68· !20 !!19{8 9i! !!1968 120

. _____________ !fl.6t_ to öl ____ , __ tt.i _________ ..!.!la. _________ !t6.t. 5!.2-- -----3.l.tJl ___ ~-- ....... a... .. . '''' i4-4D ~775 432: seao o ••t~ &i6 192o 144· !975 ,, •.

-----------~------ .. !?71. 1ÓI6·. ..J~--------~1!.2:_ ..... S.$'1- -·- C- .. ··-----!9-2-1-· -i248. --1071. ·. 3~6 -- ---1'172 192 '*22 768 eé1a· o 19aa 48o '''2 a14

. !1.773.. . .43!: . suJ. --~ '•~-~: . t&o. . .. ..Jo.aJ. 1'-l· . ,,,,s au,-'''4 1.1~4- 1124 48 !5874· 768 1924 614· !J014 '''

·--·-·------·--···- --------------·--------- ----·------------·- .!!1'1.1!L lti!S -----·---- iD-08.. Jf-?-1 -····24-S.---···-----5-9-1-IJ.. -i~Q.t:- ·--- H1!!1 -------- -------&·~ --5776 126· 1126 672 1$?6 6?2 !5926 36à 1076 600

_.!!11.7.7.-.. - ---3.1-''------------h-11---- -· --- -l60- ... ,,_,_,.:_ _______ -..S0-4---··~------5-9-1-7- 0······· ___ ,,.,., _____ ,, ____ 336- " ,.,.,8 111 eeae 288 ••?8 11~ 1928 288 191& &4ó !J.7..1g .311~ -- ... .J.I-lL. 910 ,.,,. t IJ29--- 7ó!· !J-9-,9· 1'16 1185 416 1130 480 !!18~~ 432 1930 614· I08ij 5

... ~-1!1.- _____ l.Q.!!I_6•__ - - !1.31 0- - ..... JttJ..: ______ - -6-1--4---- 1-93-1 · -- ·· - 7..0.1'----- --H-t-~ --- ··-- 1 5! I ,,82 304 5132 324 , •• ,. 72Ó !5932 118 !5082 43t

_!t?J3..3 _______ " ___ .Q.L ________ IlU 33.6-~------!!IUl-----··---384........ 1033 -· -4~ ----.. ~-U-&3:.. 0 !!1784 384- 5134 1320 Sèè4 0 !!19!4 ,,.. 9)84 331

.......... - .. J.ll..! ................ 3!.4:__ !t.U. - a&.& -- -- .S .... J 1t!'2 I.J3S· ó 898!9• 86tl· 5?86 96D 1836 264 ~-~· 1344 5936 102· 9986 384

..178'1 1•0 1837 8-1-6. . .. SUJ - -0- l-9'-1---------------113---.. 1981 - 36ó -------------·--- efaa· ·· o· - 1838- - -·-· !'J,.,--------,8~8 o !5938 211 !Jt8s 240 '189 l1,2' !'J&3t. 0 !1&19 480 1939 8AO 8989 . 31tl ---------·----------,-f()'Q- . 4SÓ· lt4o ... i4o- . -1100 5'16 ·-·---,94à--- -- 314- !J90Ö 1141

... !7.9t ó · .... -'14-1: --- -itO.o . la.o1-: 624 lt41 116 19tt · o ~~92 216· 1142 288 1802 336 1942 Ïlll !1902 192:

.. ----···---- -·-------------. JJ793 . !1.8, _____ _!l.fl---- --- 6JJ.Q . !813 · 116- -- -- -!9-9-43 Ö ·I 993 !Jc7(a.c 110• ll' 1144 336 1804 .,. 1944 a•o 1to4 ióeo

... !7.9.!!1 .169.· _________ ..!. • .45 ... _..... 38.4 - . ,.... . ·-0--------194!5 168 ·109-!5----··· .. --··-38-4 -!5796 ,,,. 5846 012 18•6· ~9* 1946 iaóe 19t6 3tt !U.9.' Uli...... ..ft-41----- .. 0- 1891 · . !1'6 · !994'1 tta· 199'1 8!6 1198 ses; 1848 102 seoa ''' 5941 a stte 431 !19.9 ..... ----~--------0--·- ______ JJ-i-49...... --- -1241-· 1-lf-9--~~-------43-2------1949·· · iOIG··· -!Jtf-9·'--- ·- ···~--··-· -18ÖÖ 168 1 11•0 1080 IJ9ÖO· 36G !59!10 6,1. 6doà 0

Page 27: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

N '4Uwt: -·--·~"~-~~·--· ~"~·--·---~--·- ·-·--·----·~----< _______ .......,..._. ___ ~···---~-~-""-·-·-·· ~<< NU~

Page 28: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

NUM

5'1~6'_~ --- ~ __ 6_llt'---'------~--'--04 __ . --~- 6Q1. -~ _ _ _JL_~- 6!1ll1.. _i!t!t!L ____ ~641JJi· .. __ 4QJ: 144· 6802 168 6312 72 6402 314- 6412 104·

6a5l. 36D- •m -~--JL__ --'~ts J64___ _ __ .. &4.53 . a~o ~-- ~•••-3- -- .... --- ,,,, -~--------------------~----------~--~-----6~5-4 ---------~~-~6~a·o~~-~---.-,o4~-~---- 120 •s••· 1440 64o4 ~'~ 6414· 624·

-~AH!.. a . 61.05. _ 1112 6iiJ~ 314 64~5 ''a' 6415 o 6256 0 6806 960 6$16 624 6.$06 !04· 6416 336 ~~'-'- ~:12:--~-- --~ 6JtD?__ ~192.c__ _6SÎ7'__ ~ __ tea~.. . 64Ö1 _____ a~ .64.17 432 6258 48D 6808 336 6SII 456 6t$Ö8 41ó 6418 641

____ .625.~L ___ 2~0__ _ __ 6&09__ 1114 __ .. 6Slf -lil- 64-ÖI . 5'6 . 6.4JO- - 7-!Ö 6260 336- 6810 116 61~0 314 64ÎO Ï344 646Ö . Ö

.. . __ ft6L.. 9..6_D _ .6.!.11: .. ..o. 6-lAt: .l!4 . 6AU !176, -- 644-l 11!J2 6262 288 6812 102 63-2 601 64~2 144 6462 7!Q

_______ , ____________________ -------------------------·····-----~'----=--~ ----------------lL--__ ...!I~~L ____________ 43:2 ________ 6:1_1_!-- ________ 4_3...2 _____ --- _6_4.t3_ ____ __ ..9:16._ :.64U. __ ------0 ' '' 6114 10!6 ,,,4 0 64,4 118 6464 168

-----~' 2 : ________ ill!l_ _ _____ __.3_~---------•~tt--- ____ _1_~·----------"~~------ -----------~~----------'_.-'!~~----- -----1 '-'-'-1008- 6816 216 6356 1108 64t6 140 6466 4!1• - 2.~~!_- . - 6.!.tl __ --- --- J)_6____ - - > &.j!'l. J) 644.7. -- .11t4 . 6461' 48Ö

6268 o 6518 614 6S&I 241 6411 431 6468 336 ___ ------~---~~~ ___ -~--~ ---~ ____ ----~~-~~-~-!---------"j~_a_; ____ ~ __ All_f~----~-- ____ o -----~---'lft-:. _________ l6j ___ - ___ .6Ji.9_ ----·-- ______ a.«tó.______ ______ -----'•-6! _ •o.a ... _

6270 76!· 6110 192· 6310 432 6410 192 6470 1104· 6211 j _ __ __ 6!a1 __________ 100JL_____ _6S;.1 ________ 164 _64.!.1_ !104 647i,__ o _ 62:-?2 - - -108-- -- --.-Î22 432 &3;2- t02 6412 936 6412 144·

.. 121.3. ~$D 6_j_l.3 JJQ4.. . .. 6Si3 '''. A4l3. o, 64!13: lO!J6· 6274 4&o ••a• 48o '*'' o84 6414 336 6474 tt'•

- 62Z!J S~Jl:.. ____ 6UJ. . .. 3.36 .... __ _qjJ: 64U Ï344· 647§ - AIÖ-6276 336 6!16 1212 63;6 408 6416 l1,!' 6416 36á 6.271 ~2: _,,., - · 0 "-f' -7c68 6417- 2l6- ,,,,:_ -- 410-6278 1104· 6128 1t2 6S'a 432 6418 a 6478 ''' 6219 o; . . 6119 .,,. - -ü~t. 4Q& ••tt---- · ,.. -6-419 o 6285 240 6830 576 6$10 114 6430 431 641Ö 216 .62-!t 91!; ______ 6131· 432-- - 63-it ---600 ---6431- I) - 6411·-- 6-'11 6282 936 6832 ó 6ièe 432 6432 toa 6482 ,,.

_____ ----------~---- .62.8:1 _________________ a.e_sj__ ______ ,f33--- -- --- --4-BO- ---- --- 6--$13----------- -o ----'4-13. --------------2-11-------- - 6413- -- --- ~a sa -- -6284 4~! 6534 480 6384 0 6434 t44Ó 6484 2t6 .uas ..... -------A-Bi---- --- ----14"----- -- --- - o------- •eu.~ :s.a•- 64!! 4-8Ö - 64t&· 12oö 628 516. 6136 72 6316 1344 6436 336 6416 168

. ___ _ ------~-- ---- .. -U37. ~- ---- ----3-U- ------ ------63-11--------------U-2------ --6--4-3--L-- - ---~-~--8-16,-~~-- --6-4-1 '1- - -- -- --- -~ -·· 0-6288 144· 6Î38 720 6318 ~68 ~438 516 6488 360

- ---- ---- -- _ _ ___ 6.28 9 __ -- ____ i_tt ________ !t.:Ji_ ---· -- 4JO.. 6-$ ~~------ · -~...!JIÓ---~------ -6-43 f---····- - ------0--- ~- ---641-0-c ..... -- -0-6-0- - -··· 6296 ,,, 6840 288 63~0 1l52 6440 384 6406 314

... 6.291 9.eó. . . 6ï•t ... u•• . . •sP.t o 6441 ,,._ •••i· '"4· 6291 264- 6!42 576 6!01 314 6442 !60 6402 D

...... 6.293. 67 .. 2 ____ 6.3 .. 43 o. •s-.s: sas 6443 614, 6-4t~ .aas 6294 8!6· 6844 528 6304 ,,, 6444 360 6404 1151• 629!5 . OL 6!43 11!12 _ _ 630!j 3.14 64•S .. 6'1 - 640!5 Ó 6296 641 6846 480 6306 0 6446 !104· 6406 101•

. •·· .. 6.2.9 .. '2 4:!2: 6,.,_ 672. 63t'1 600 644'1 0 641'1 ,,. 6298 !!S= 6~48 200 6!08 '68 6448 48 6408 614

_ 6299........ tása:.._ •••• ---!1.6.---. --6soo o 6449 o6o -- •••• 336-6355 Ó 61!10 960 64ÖO 30 6410 671 6!1àii 516

Page 29: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

.. .NUM. ·N. ...

~-~ .. !S:b.t tts~: . &151: .. ·--·-·- o 6611. ' '' 1111 6.oo 67oi --- 936. ······-·--~-~-------- -- -~-- ··-- ··· 6so2 -----~-4os. ~~~2 432 &&o2 6oe 66!!J2 o 1102 sa a

6503 t .. 6f!13 --~~Jt. _ 6613 2,8 66.!HL.. 84.0. . ... 670.3 .... 0. 6so4 z-o 6~!14 too8 6&54 1t2 66!!J4 614 &7o4 1!6 6SO!J 384· 6i!IS ~-~~ 66àJ 164 6.6.13 0 . 670!5' 160 6566' . .. 11~16· 6156 0 6666 1440 66!16 72' 6706 671:

·-·-···········-·---~---·-- .................. ~~g!.... ~8~· ~·""-- . ''-'- ........ .. ..... ~., 0. ... 66!!11.. ,,,._ 610'1 672: 6!508 168· 6118 710 66Ö8 240 6688 336· 6708 191'

. ····--···-······--------~-- . 6'Q-!. . ........ ~.&!LO.=~----- ... 6JI.i.... . ........ 0.. . ..... AÜ tL ..... 1.28 - 6U9. .. ..... S!ll• . -6709- --7fl· 6!1J10 ,,, 6~60 !92 ,.,. 336 6660 .,,, 67i6 .,. 6.!U.i 0... . .6161 .tlfL .... A.lS..S.. 10.Ö.& 6664. 111' · 6711· · Q· 6512 0 6162 !176 68[2 ''2 6662 [08ó 6712 240

--·~--------·--·············-~-····~····-·--'-~-~---·-····---'.!.~L------·-~'-U.; ___ ~ __ .6!A3...... . ... JJI2 .............. ---'.6i~~- ···-~ 410.--- ... ---6663 ·········· ·· --·---0- ·· -671! ---·-- · -··64·· 6514 48Q! ,,64 128 66~4 1464 6664 336 6714 936

................ - ... - ............... -------· .. --·-··--·-----'!li.5 ... ___ __i!_!:__ ____ .... 6jJ_!__ _ ......... .5.76__ ---- ... 6UJ;..._ .. -.... ..... -.0. . --------6.648-"---- ... ·· {!l.JI:- - -· -67-15·- .. ·-·- -----1-88 ···········- ........... . 6Si6 600' 6166 1!12 66[6 264 6666 768 6716 0

.!.J:i'l-- ____ J!_,,___ . A!.A! -~--~ _,.ij;'1-- ·- 6'2 666,·- · 1·14-t' 67tf Hl 6s1a '~4· 6168 11e ••t• 768 6668 ~ta· 6118 486

___ .. ____ .. ___ ···----··----6.5.1~9----- ~-~--iL-~-----•u•-------- ... tJ84- ----...U-•--------3!2 -·--·- ·-'•'' -- --· -it.-H·-·· . -·- ''t•- . -·o 6525 !36· 617o 1tti 6•jo ó 6&to 410 6716 06 652i ... 8!..6~. ... ______ .AI'-1---- - 36.0 . 6-611· · -· --1!12: 66t1· 8· -6111 48ö 6522 624 6172 240 6fi2 114 6612 144 6712 768 6523 336· 61173 41o •+t! . e 6613 336 &113 &48 6524 a &174 624 6f94 240 6614· 1248 6714 146

. 6!2! 1.080.: .. 6!'15 . 0 .. ,. 176 ,,,, 316· 6?15· tltÖ: 6526 528 ,,,6 144 6616 1t52 6676 3!.2 6716 1&8 6.!27 j_ _6J!7 ... 432 ...... , ''2- ,,,, . Ï0-16· ,,,, 0 6528 49 6,78 1016 6ti8 t02 6678 410 6118 ,,.,

.. 6529 96,0 ,,,, . 864 ...... .9.' 144f 6-619 t 6?ttJ. 6!4 6,30 !488 6~10 102 6630 576 6680 336 '''Ö 614 6531. !!!&...... . ... . 6.l8i ... l3.6 ..••• 1 ·0 -6611 060 6131 --816 6532 192: 6512 4!2 6632 104 6612 !88 6732 240

.6.53.:t . 3!.L --6.183. . . _.() ,,,,. '2.0 6643-- . -· ... '''~ 6713 ~-.. - .• ,, &534 tos& &984 518 &634 164 6614 .D 6134 t44ó

.............. 61.3!....... . . &iU. .7.6.8 ...... _. . 4.6:J.J. . '10 4615 61t· 6735 ri 6536 168; 6i•• 624 ••s• toa 6686 a••· 6736 1aó

_6JS.31 .... !..LI· ____ ... UI'l __ ·-- -~--6.!4-----~---.....U.;·___ -26-4--- ---·---661'1-~-·-- --0~- . c-'·'-1·1- -· --9!1"' .. 6538 336 6988 ó 6A~8 !528 6688 ,, 6738 ,,, 6539. ... 6l2.:. ............ ~ .. 6J89. . ... 5.2.8 ... ,,JO.:.. --~- --- ....... , •••. . ....... ,..,, 4!1 654o 192· 6Jto 912 66•o o 66oo 864· 674o 6!4, 654i- 672~ 61.91- Q 664!. 144! 66t1· ·5Ö4, 4?41· 1UJI· 6542 12Ó 6192 0 6442 1!.6 6602 314· 6142 311' 6643 ~ .. Q.... 6tf3 . 11.6 - -664:J.:~-- . 218- 6603 ·38-4, -· '''''- f) 6544 192· 6194 6?2 6644 516 6604 696 aj44 511 654!1 76!· . 689! - 384 . 661!' - 5'1-6 - 6-605 D 6?4!9- · -·· 384 6546 6'12' 6196 !16 66Á6 914 6606 5?6 6?46 1214 6541 . 2.64! 619'1 140 · · · 6647 G 66t'1 534 ,,,; 384 6548 456 6198 312 6641 240 6608 Î2ÓO 6148 0

.. 6!14.9 ... 768 . 6,.,_ 0 . 66ÁO . 612 6609 5J6·- 6149 1008 6555 5ft4· 6loo 41à 66ÎO 1208 6?öo t68 671Ö 1111

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JtUM NUM

6' 5t· . ·-----·--~- ________ _6_f1l1:_ ____ ._~_,lo ... _ _ __ tft1.: _ ~-~--- _____ 6tnt__ -'•• _ _ __6H!r~---· ----------~--------------------~---------,,-,-a-~ 120 6eoa 4ao ••ea 412 6952 6'?a· 6912 191

6153 _____ ?~_O _____ lt0..~--~--41~--- 68l3. _;514 6903.~-- tL. ______ 69.13_ 7!1 ----6-754____ 511§ 6104 624 6884 1!04 6904· 218; 6914 11!1!•

t;;~ t~l ::~~ 1~:: ::l:· 24: ::~: ij!:: :::: :::: 6'1!51 !84• ...... 68QJ . . 0 _68l1 1_18 69Q'1 4011 69!'1.. -- 9!Uc

---------~--~---------------- -------6,58- · -9&o- -- -&äo8- - ·- a4o 6$ë8 ''6 69öa ti 69!!J8 431

6 '159 .t --- ___ 610 9 148tt_ 6àlt___ _41_0 69Ö9_ 864 .. 69!9 0 - ---6-?6o 312: •11o ''' 6••c 114 69to •sa· 696o 96

___ tz_~~t__ .... ~-~~4~-- _6h.1 504.. 6.8A1. 11.6 .. 69.11 Ó. . ..6 .. 961 614· 6'162 612: 6112 0 6$~2 188 6912 32 6062 '108

_______________________ ---~----------- ______ 67:6_:5___ _ _______ 20--------' 11~--------'JBL _ _ ____ 6à.6.l; ______ ~ ___ o _____________ .611.3 -------····· . .2.88:. ------69.63-- .. . -----------!88-6764 432· 6114 !116 68,4· t02 6014 {296 6964 Stl: ---------------------------------------- --:-l:t--------!:: · ----- ::i:· ------ia:---------:U:-~---------i~~t--------!~1;--- -----~ :::~------------::::.--~-------·tt:t; _______ _

.c6'lA.7..... ..Q.L_____ .ili1..... 3.8~L-- &t-6l.. 4.80 6tl7 600 6«)61' 0 6768 0 6111 672 6868 t92 6918 410 6068 118

----------------------- -- --~~:t~f----- ---lit------------:-H-~------- -----IJ:-·- ----~--:tt:~--------:~:--------~-tti~~- .. :.:- -::t:-··· ----::g-~- -.... 611L .. .. 31U-'--~----6.At1----·---~-----1Af2----·---6·~1----- 0~ 601~- tlótl- 6.011 t08Q

677a a64· &aaa 1t2 '''2 s!la 6tta s•o •o'a o 47.1.3 8.64 6&13 . 0 ,,;,. 672 6l13- 384 -6013: 336i-6t74 J6Ó 6124 . 696 66~4· 768 6914 102 69,4 !ÓOI .6l'1!L- . -----4- - --~--Ui!l -----672 -6à_;J__ 600 6915 IÖ4 -- 6976 0 · 67'16 624· 6126 600 ,.,, 0 . 6916 Ï680 ,,,, 71 677, - ----- -5-'-'-- -'-l-11~ -408 ••. ;., - -- 60-8 . ''"' 0 -697? 6!4 67?a 360 6128 140 61'!• 720 69!8 144 6978 ,,, 6.1'12 . ''-''- 6119 -504 6 19 > 0 6919- t4~0 6:97-t 3:16· 6780 0 6130 1104 6880 !44 6930 U.!ll: 6910 480

---·-----····- ___ __ _ __ ~--- ______ 67Bi... .7.44: 6.13-1· 0 6881: 113-t ~69~1 . --211- -6911 768: 6782 1248: 6832 48 ·-·~ 480 6932 4ól 6982 408 ... _______ __ _____ __ ------- ________________ &..tal. ______ ~ ______ JL ____________ .6.813~------ _u.a ____ ~ ··--~ ,._.,_ ___ . --216-- - ... &o:t3 ... sta.- -- - -·· . 6083 - .. q

____________ :~~:___ _ ,.Q;_r _________ !:l: ____ ~- ___ __:;~. ::::: :::. . :::; '': :::;· :~: 678

' ~It _________ !~~1 ___________ I~_: -~--!ttl _____________ ~~;--~-------::~~---·--------~!:t _________ J::t_ ~-----:=::; ____ _ -----------------~----,--··,--_~ a···a···--------- 336, 6Atse 576 6Î81 aas 6938 914- 6t8è 126

--···~------ ___ ... 678_9 ...... .8.6~_. _______ 613.9~--- -- ----O-------Ai~0--~---~--1-1-0-----------4939-~--- -·f'1-6c----~---6N9----- 11 .. 1'-· 6190 48Ó 6140 240 6ÎOÖ 576 6940 0 6000 5;6 6?9i .. ..Q· 6l41 614 61Jl- 614 6941 Oit · 690t Ó 6192 10~ 6142 1200 6802 SlO 6942 ''' 6992 240

.. 6?9=« 4.32~ 6143 _ 2aa •»s:. 1a4a 6943 n ''-'3 ,,.! '794 516 &a44- ó 6894 144ó 6944 a es. 6994 4:!1

_ 6'l_9!J .. 564• 61.45. .. .J.36. 68J5· -Ö - ·· 69.t!c · 611- 6005 - 161 6196 264 6146 672 6806 0 6946 384 6996 480

. .6.'l.2.1 85.6 6141 0 ,,,,. 111 694, 606· ' ''' 3!1• 6798 5'6· 6148 '12 6808 480 6948 240 6908 936·

·- 6799 --- - - 0 _ ..... , ~ - -S.44Q. 61!-9 ... - 840 --6949 . - .. 84Ó . ,,,, ti 68ÖÖ :s:u, 61110 720 6900. 480 69!10 !176 7000 281

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NUM

-----------~----io_at-~----------''2: ,,_,1_:__ -3•• ... ;iöt~ 1,,, _ 1111 o --- --'2-0-1"- - ·~6 '~ö2 e•o 7~52 288 11§2 432 1112 o 7aoa 8t6

____________________ 1nla 384 1113 48[L _____ _ 7.s.o3" o 71.1~- - --«.& - - -'2~3 aot· ~tia• o 7&54 624 'i64 o ,,~. 1344 12ó• 336 700!1 168- . ,,,, __ Q 710.!: 672 71-l! 3-84- '1205 ,,.

-- --- --- 1~1~ 96 ~ - ~ti;___ _:~~--- -;_f~l ::t_ ~i:; i~il. - ;:~; 96 ~ _________ ;[~_: ___________ ,:t ________ ~_u: _____ I:t ________ ;f~l- j:~ -- ~t:t --~~=- - Jil!-- 1:::~

_ 'QiQ &OD8- 7f60 240 7iiö 960 7160 614- 721ij !U4 ... . - --- ... .7-!)~1 . 720 . 1.J61: .. . .. 1440. '-tl1'- . 0 7161 .,,_ 1!Î1· 840

7g12 240 7162 576 ?t&2' 480 7162 360 72t2 101· __________________ ------------------------------------'--0-13---------~-'-1ll·-----ma_ ___ ~---------O------------'t.'-3~-- ...... sa- --- --'s.u------ ---- -«o~--------1-a-t& ------------ : s-o-•··

7014 960 7164 600 7!14 702 7164 0 7214 1296 ____________ _1.iii! __________ L_ _______ ll6.!L ____________ ~f1...6 _________ ,fiJ~---------.:.HD----------'-1-61•---- - ----3-U-----------'1-2-l-S-'-------------i-----

--------------------------- ?ói6 486- 7j66 504 +!t6 240 7166 ÏB36 72{6 ,_.._ •.

7.Öi7 zan~ ---'-f-6-'- -------33.6- - *i-i' -4-12 -7-!;--6-, - 0- 791-' _,,_ •. ?ii8 112: 7j6e o tii8· 816 7168 _o 12~e 'la

···---------7-0-19----- ---l.o-!.2~--·· ·----1-1-U.- ------· -61-6-- - ·---~-'1-t-!f------------ -- ---9-------- __ _,,__,9-·--- ---- -·t-tJ-"----·---'-1-t--1 ,._ --- ··--···· . !6-Ö- - . 7Q2Q 384· 7i7o 1112 tièo '' '1'0 ''' 1eeö 5D4 '102! 6-~2 7f71 -D 7t-t1 12-•Ul 7t11- 48~----- 7211 671' .. -. 1if22--· -----Stl: '1Ö?2 ____________ 96 _____ -,.~2---~---- s76- 7172 43, 72t2 !JIU

"öa:s o '''3 1ooa '~•~ m•tt '"':s ioeo 7!t3· o 7Q24 .o 7f74 768 1$14 816 '1~74 71C 7!24 480 'Q2! 1200·- . 1-~7!L- - 50-4 . ~tP' 1!12- '"'5 o 7t•!J - !10 7026 960 ,.,, 768 ,,,6 432- 7176 384- 7!10 1114 7gi' . tAf,_ ___,,,, .. 3-84 -1l-t': a 1111 .. 3!4- 7tt7 7tQ 7028 33~· ,,,, 312 7t18 408 7178 t008 7218 0 7Q29 &1-40 7f79 - Q- · - ?t.9' H4- 71'19- 118· 1!1! 1'04-7p3Ö 384· , •• 0 480 1~30 864 7110 .. 1~·· 7230 '''

. 7Q31 --·-0- --'fl-1 .... - 768 ··- 1l~l' -- 48q 7181 !200 723t·-- . ó 7g3a a•o 7f82 936 ~l~t o 1112 ''' '1232-· 96· ~1i~ ,:!:-:- ---·jJ:t· - --------:g-g.·----·---· ·lt1~-- .... ~::· ;t:~ ~-·; - --- ;:::~ .. -:::: !D-~'--- 3.~t _ _1fS.!:_ . 86.4.. 'l~l' . a 7185 96ó '12SJ Jll· 7036 0 7186 720 7!16 384 7186 431· 7236 ,,, 1j.3J_ ___ . ____ _Ij 4· _________ 'li8.1 -------····------··--0~- ··------1-l.$t·-··-······---- 484--- --------11-U----- - -----60~······--·--~--··· '1»1--------------JJ-t·---7038 !64· 7t88 120 7!18 672 7188 288 72-38 864-

__ ,_03_9 ... ----~----l ...... -------'lll.i t-1!12. --- Jist --- -936--- '1-119- -6-12-·- -7a-Jo.._ o - -· ~545 144· 7jlo 614 'i~ö 384 '''o 1112 7!45 a•d 7g-tt. 624· 7jl1 112- 'i~t- '?IQ 7111 0 724t' tBS~ '1o42 !r&· 7812 36o ?142 6ao 1102 a•o 12•a ••a 7o4t~ 5fU:_ 7113. 2&8 ~1•3 . ö 7tt3 ••o· -7243' -stl· 1o44 a•o, 7it4 16!16 '~•· a•o 1104 1•• 7244 5!a· '10 4!1 4~0 7695 ó ,, •• -- 1!12 7199 384-· 7!4!5' t&ó 7o46 1200 7il6 246 'i•• 1112 7116 o 7246 sas

. 1.041 0 7197 1148 ~i47 336 7101 3f4- 7241' ó 'Ö48 289 7198 624- 7i48 ••• 7108 336 7248 144 tÓ-40 11.!12: _,,.. 336 ti 49 116 7tt9 à '12-49 - 518 1o5ö 672: 'Îoo a 1ïia '1a 11oo 211 1216 t••ó

Page 32: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

)lf~~~. .. ....... N.. ..... JiUM.... . ......... N Jf.UM .. Ji.. .. N.UM. -~ NUM

12 8!1.6· 1101 1112 7~11: a 1451 __ ,ä,j ____ ~~ ~--~'-4l1 ___ ~- .84Jt.

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1254 tó~e. 1S04 612 1314 648 14ö4 336 14!4 144Ö ...... 125.5 0. .. 1!03~ 480 ?I.!: 864 '!40!5 671~ 74.1 ó

7256 552: 1106 912 ,3,6 0 14Ö6 i104· 1416 96 _ ... 1e1 . _S~4·~--~---.110?....... 60!L...... 1.!!1· ~ !ltL ~1A.Ö1 .o ~~ .. l-411- 816-

1258 432· 1S08 288 1!is 120 1458 ó 1418 168 ··~~-··-----·····---··~--~~---·-~~--·-·~··~~---~~--~---· -~~--~~~-·-····· '23! ... J564: ........ ~'lot_ ..... 6o.o '-P' ··~ ·--~ ~ .. .e- ·-· .. ,.a. i4'-a· .. '4'~ ~ aao

?26Q o 1ss.o tol6 '''a 48 14to s'' 146Ö ''' -~-12.61. .4..!!_2_: ....... 'lAS.-1........ .o. iU1 13.A4 1411 6-ób.... ---~'14-61 1Sttl

?262 564· 1$12 96 ,~.2 1!08 14i2 431: 1462 ~80 ········-··---··~---~·····---·-----·-···~· ~--··~·····-····---·'-·2~6.3.~. .. ..... _. __ Q _____ ?.U.~---- ........ 61.2~- --~----~~U-~----------..2.U.~---~-~"•i3. ··~- .. 61a: -·-·. uu: ..

1264 1·8· 1114 1112· 1364 410 1414 518 1464 384

----·-------~--~----------~------- ---~t~::·---·----1 :: :-: ··---~-:t!·-----·· -·!tri~·----·-····~-·tit:·'--·-··· ·····---:-:~·-···--··-··--~~·~f~~-········~···-~~-~,.:-; ... ~------~:::L--- --- t::r··--······ -· .. ~. .... .. . .%261... . .. ~.31~.. .711.1... J.!l.tL...... .fJ •. 7 . -0 .'741;1 188 146!7 · · 480 ~

1268 480 7J18 504 ,,18 tOl 14!8 744 1468 120 ~---·····~---·---···--~~;}~~- ·-··········~·--ill:---·~-~·J{ll--·-···-··iai-·--··--·--·;Jtl-----····i:~: .. -·----j:j~~··········· ······· · tu:·~······ ·· ~-~~:t~~ .. ~ ... ·· ~ · ~ 1,;# ··

12 ~{ . . . ~ Q. .... Jiil ...... ~~- ..... 6.'?a ·~ .. 1:~t1. _I_11L. . 7 4J!1. . .t2ÖO . 141{ 0 7212 JiO 7822 816 1112· t44 7422 614· 1412 161 1213 480 7SI3 432 1S'3 528 '7423 0 1413 384 1'214 a 4 o · 1 Î 2 4 o 1 si 4 1 t o 4 '4 a 4 12 1 • 14 '1 4

. 'l~2'!!S ~·· ..... !80_ ~~~··-~·til !I -.- .. J.,12 .... 1J.!!t.... 0 74!5 1~-········· 141!5· 960 1216 240 7126 710 1~16 160 '1426 168· 1416 480 121.1 -~6.0· .7l21 .a 1Sl1' 120 7417 671 14'11 S6ó ,~2-?e · ets 1s2a 312 ,s;s 384 1428 a•o 1418 &4o 127 .. 9 á .· 7.829 1016 1~;, 672 7419 410 '14'19 0 1280 0· 1830 432 1!10 488 74!0 i104 '141Ö 19t 12.a! .J . .tJa·. ___ . ~'-8.31 ..... ~ .... 112 ,,,,_. ..,ta . _,...,,_ ... o.. ~ 1411~· s.s•a· 12ea aea. 7332 21e 1112 792 74!2 _a•o 7412· ''!

.. 12.81 .... ~--~ .. ADllL~----·.1!33.~.-- -~408 .. ~·--~.~7383.~ .................... o .... -1433 ... ···········!-US.· --1413 1~0 1284 432: . 78iU 11112 7if4 336 14!4 110 1484· Ö

.......... 7..aSJL .UA., ........ 2tn. ... .. .. . o '*'' .. ·~ •••· '"·'' .Q40 14-85· 116 1286 lÓ8Q 7136 432 1$86 614 7436 360 1486· 81.

~-~-.. ······------·-~-·--··· ·-·~····-··~- ... ~ .................. _?.2.8-' .. _J, --~----~--"-111 ... --------.'16.8 ····-···-'"'-·---···~~-----~ ~31-----~·-----168- .. - ......... , ... ,.:. .................... Q. ...... ·~· .. 1288 1~2' 7~38 384 11.8 .. 0 7418 336 7488 12Ö '1289. ..14..4.QL .. ~ .. --.1l3.9 .. ~...... ~-432 7JÁ9--· --·-10IÖ----------- -'439--···-- ·· · ... ·9- ~ 148f - ~ ·~~ ·864·· ?295 12,2: 714o 240 1:soó •24 ?44ó 10i ?4t6 t6ó ?291. aas. 7141· .. 16&. ?:sts.: o '4"1 ••o ?4t1· 314 ?292 D ?842 624 '391 96 1442 ''' 7492 144 ?29.3. !!16 ....... ?JjJ . . 0 1JOS :184 -'1443- IÖ. ?493: 01Î 1294 3?6 7544 192 1304 1392 1444 416 ,... 611

. ?2~5 . 0: ___ ?l4JL 3114 130.1' . . 3!4 .... 1445 .. 161 ?49!J· ö ·. ?296 96· 1846 1536 ,306 2'70 1446· 1248 7496 611• ?291 511: rs•? 314 . ~311· ''' 14'' . o ,,,~. tooa-12'9'8 6!12; 7e-8 288 13os t6o '1448 504· '749e 480 ?299 140 .... '~·· .. 1.310 ___ ... 1319.. .0 .... ~ .. 1449 768 .. 7409. 701 · 13öo· ~· 33·6· -- ~---.,-,,o .... ··· 84o '4öo •oo ''!lo sö•· ?soa 296

Page 33: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

. _ .... H ... .. .NU!L . NU~

····--··-········--------·-··--·--·- --·-------- . __ js_Q_l _______ z~o· ... ?'.15.1....... ___ a. l._Ö1- 16to 7611 .. 41o. ''~i·- .67'2-· 1soa 10S6 1~12 12 1•ga 4&o 7612 432 71ó2 408· 7501 . 0 7tl3· . SA~ "!6.03 2.64 14.13 .... . .. --.6U' ''Ö-3- ó -·--·---- ------,s-o-. 192: 'f'• s64 ''ö•- so4 7614 ,... .,,o4 :s•o ''ö' ;~ó. 11f5' 2•$ ."! ___ 6.ii!f_ . 1eae . 76.!1.! o 11óe· ''' ;·5o6 7& 9 7Ï56 864 76Q6 4~6 7616 176 77ó6 UJ11' ·····-····-······-·-·····-····-------·-·-·-· ··-~=-l~- :~:: ··-·-·-----~l:i- . -:~: l:g:· .3:·· ~:::· i::: -~~~:·. 3':·

---;;~~ -:;~:-- l:~~~--- ·· :sa~· ~tf~- · :::- ~:!: i!: - ~;:: ~~::: .'l5Ü.· Q 7.1.61. . .. 4.8.0 . '•lt· 480 7661 . {216 '1711 0. 1512 288 7162 720 76ÏI 0 1662 431 77{2· 240

--···-·-·--··---··-·--··-·--·--·--------------···---··---·---?.!1.3 ...... __ -----~:S.l.:.... _____ l.JJ_J ________ !'l.6~.--- .... !.6iJI .. ···-·····-··--1'!6 ..... --······--'-663 ..... ---7'1!3-- . -U,!II; .. -''14 1aa4· ,,,. a4o 7&1• 1•:se 7664 o 111• 486

··---····-·· ·---·--···-------------~- ···--·----"!.5_15 _______ , ... ~----'-!Q..:_ __ ~ __ li6!_ ________ .-1!.16. .. ······----'liS---·-··-····· --- --0---_____ , .... ___ ·······''' --- -- __ .,.,1!1-'-···-- ····---4! t:· ......... . 7!Jj,6 . 0 7166 864 7fÏ6 0 7666 ,,. 7716 336

. .. •. 1!l1l 11!!; .!J..6.7. ---0-- ... '"' . 432 .,,,, ·48-G,· '171,.. .. . ,,.~ . 1518 &72: '''e 144 76ie sa8 7668 sa• 7718 &16

--~·-~-------- ·---~~----·-·-·--~-~---·-·-·-----... -·~------·--~--~U~i-O..-----------...o.:.... ___ __,J-6.9.-------.aJo ... -............ ,.~L.. .......... 1t-n-•-----·-'1669·-~--···----••o--·--.. -----17-tt .. ·----- .. ·----- - ~--~--·· · 752Q 240 1$10 720 16~0 288 1610 it8. 711Ö 240

· -··· 1121-·- ... ..j64L ·--·----1"1--------·····--· 4..12--· - -'?·&f-1--··--···--····- '1-9!~----- ······--·· '16 11 (!······-········ · ...... 11tt-···· ·tS44 7522 aas 1112 240 76it· 11~0 7612 111 1112 168·

.. 1523- 8AO. 1-1'3: 4S6 ,61-3 0 16'13 61! '1,13 2!6 1524 720 7174 1440 ,,.... 240 7614 lt6 ,,.. 504·

··- '1525.. _48Jli - .... ,,,,_ .. ~-- -0- . ".,.. ~1!12- .. 16,, !504· '"'' 110· 7526 l296 71'16 360 1616 672 7676 0 7116 816 ;sa% __ --tL. _,._,_, . .&1.6. . ?6f'1~· ~-2-4&-~ '16~7 6óo '''' o 7528 216· 7178 936 7618 :S12 7618 431' 7118 96 75..2.9 72.0. 7.119 ... Sl4 ,..._,, 1tJ2 7679 -f- · 7119 5!8 7'535 &64· 798o o lf6io 480 76ao 48 71:tii &!6 153.i s~•·- ~---118..1.~~--- - .. t12- - ,,., .. _ -- --o- '1681 111- - -1111-- .a•o

.. l~~: __ .. ... _:~::_ _ ____ J::t __________ ~~t ... ~~-- -~t~i. --=~:~------- ~::t ............. :::. . .............. :;:r . =~: 7534 432: 1184 240 7634 1440 7684 480 1134 1104-

.~!3.!5 l;.... 7!15. .. .384 ;u. . 48Ó 7611 i til 1',135 0 j536 0 7116 1686 ,.~. 336 7686 1584 7t36 614 753i .su .. , ........... ~.7Jt1..~- .. -- .. --!1..6-c-----------'1·i31--.. ----- .... Jte---------'1-617· -- .. ·---o~ ....... - -11-31- - ·- 4!t' 7538 ?20: 7t88 192 1.&38 384 7688 306 7138 431

-- 1!39. ..... -·· . .!1 .. 2~- ---- 'l.tJ.Q ______ .. S.-464 ...... ·'1i!t-'- - .... ---0--- ..... '1689 ........ ... .. 96ê .. --7'1!-9~--- - . 960. 1540 192- 7190 576 164ó 672 7610 314- 714Ö 6 '15A1 . 1..1!6 7Ji-1 0 '641 ~ 1344- '1601 tG31· '1141· 696 154t !4Ó 7192 336 1642 408 7602 240 '1'142 864 '1!l4 .. :S 0~- .. 7t93 ... 614 , •• :J.. 606 7603 tó•- ... __ ,,.! 0 '1!44 7&8, 7t94 606 ,,.. 0 '1614 9tl 7144 78

_ 7S4!J &...&4· ,,., 164 _,,;.,: 314 '161!· a ,.,., t2l48 7!46 768 7t96 360 ,,~, 96Ó 7696 101 7146 ,, •. '1541 UÖ 1f9'1 314· ?64'1 0 7617 9tt~ 714'1 384-7!548 0 1198 1008 ,,.8 96 7608 480 ,,.8 ''' '1549 840 7t90 0 ,649 1392 760-9 648 114t tt!JI' '1!55Ö S o o a: 1 ID o 120 '6!ó 120 g 710 o 6'12· 111Ö 516

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--·-·· --N.UM-·-······ N NiJM-·

·--~----------------------~------~---~---~fgt--- ------Ist- ---;:gf---~- --:-::-- -;ttr- --:::---- ----~:~t-~ ----- -Iti~~---- -~:H~ 2at ~--------~----!'~~-- _ _ 336- 7803 608 !lil, _ 1.212 .. _7tl3 _ n 111.1. •aO. ..

------~-------------- 11s4 -"-,ös.·-~----,-io4--------- ---o--- - 1••• ,,ó 1954 :ss• ,,,4 ,,,_ _ .'-.!.5.5 !.lH- ... 110.5: 1152 '1U.I: G 195! . !576 - 7-t!JS· 1016-

1756 1,!: 7106 624 ~.,, 216 79§6 864· 7916 ,,, ------------------------~-------'1-~1 jlQ. ___ ___ 1in1_ . __ o_ . 1U1... .at.6 _ ''-«'----. .löA '"'- ''''

-------------mi ::~ ---m~- :n- nu m ~m m ~m ::! ________ 11.6i J,1,_2: .. l.lll~---- ...... 10.0.8 '-8it -'.24 79-11- - -0 - ----1l6t: ttl-

7762 624· 7812 480 1661 936 7912 144 7962 121-----------------------------~-------~-~-~---~--1Z.~_3 ________ _,_!_8 _________ Z.l".J __________ .2J~------ ____ t8_6! ___ -----~------·····1L ..... _______ 11.11t___ 864 _______ 1.1U:.. ..311: __ _

7164 aea 7114 t84 ti~• 432 79i4 864· 1964 . o -~-----·--------------------------~---- ____ 116.1 _________ --~ti-_~----!~-~~------·- _____ __n-'--------- .!~ft ______ U!IA _____ .. 1.91!-- -- _ ..... 33.6 ..... -------'1.9-'-!.'------- -- -t-l.!JI:. -

1766 15~1- 7116 336 ,.,, 1200 1916 512' ,,,, 168 . . . - -

11':Lh-'1.. - ... -- tl-- .. !.JU __ ------ --1-liA.. - .. _,._.,_ - -264 1 9f1 768L -'1~61 - · 6 7~68 264· 7118 624 ''~8 e 79(8 431 7068 96

~------ _________ ------~~~~-- ~----~---~-"l.fiL69 _______ ~~----11,.2.:.. ___ ll1J _________ 3l~L---~------'tJ.6.f.: ___ -------1.64·---------- .10S,9.. .. --- -------······0---- ---1.06--9···--- 7 61··-- · ·

--~I~t . Á;t ______ JJif. _________ j.~::_ ____ JJÎ!...... ~~t _ ;_:~t .... -.. Js!.___ -~=_;~. ;il .,1.,2 ó 1122 sa8 +fl2 48 ,,.a ra•• '''2 ae8 11'1.3. . ~60.· 7823 0 ,,,,, 128. 7923.. 334. 7973 ~~ê' 7174 710 7824 246 ?êf4· 1440 7014 240 7974 t68d

--- - - ~;~: 3l~' -- ----t:r. =~~ ·l:~:· :::· ;:::' i::: ~:;: •• :. ''''J .l8Q__ 7.12.7 412 , • .,, .... _,.~ 7927.:. --0 - ___ ,.,,, 431• '''8 624 1eaa 102 ta+e ''a ,,.a a88 '''a sa4· ~119 576· . 712.9 . 1,12 ,.,, . _ o ,,.,_ ï••ö '''' ia••· 1180 192· 7830 768 7éÎÓ 624 7930 672 7980 191

.J1e1 ...... ~10.4· .1a:.u .. n. 1ai1 U!.2 ... '·'·'-1---- i.os-6 '-••t ,, •. 1782 480 7132 384 7èi2 12a 7932 o 1112· 1óo~

____ tZ __ 8.3...... . .. .. ~ Q.: ______ 7_1_3_3 ---· !'l6 __ ... __ tai.:s:...... . .4 .. 08.. --- 7933. . ---4!16 -- ---- . .798$: 0-?784 8!1.6· 7~34 552 ?ê$4 384 7934 i5S6 7984 . 71

........... 7.2.8.1...... .i:ü:. ..... 7.U.. 12.0------- ------1iliJ'. 48t 7931 0 7911 U!48· ?~86 ,,, 7136 . 0 ,.~. 1iÓ4 7936 0 7986 11!12

-------~-----·----~-----~------- ____________ '?_ '1Jl7 ..... &!2.:_____ __'tiJ.7 __ ~--- ... !.IJL-~- - --'1!1.1.~-----~-- ----#--------'19-31--- . ~---Ï-104·-··. ------.'1.0-1-;..-_. --- -------ll."- ---7788 1~2· 7838 1248 jj~e t44 79'8 ti2· 7t&a so•

.... . 1'1 89 . - ... Sfl4- .... -- .. 1-U-f---~ -0 ---· ·· · -'1fl-J--- ··-----1136-- -79!9--- ·· -- 480 ~'1tt9·- -8'16,· 1195 144~ 7840 168 ~~~0 176 794Ó 480 7995 486 779.i 0 . 7l4S. . - 2.~12- . .,,,.,. !88 7941 i·S.04-· 79fi ó 1192 0 7142 176 7802' 104 7942 416 7902· 176

.?793.... .!!76: .. 71.43-. - 384 . ' J.a0-.3: 6Ó~ -79-43 0 79t3- 5!1 1194 1584 .,.44 384 ;,,.. 192 7944 ,,, 7994 1631'

.. 7.1.9.5- ... 314.: .7141 . - 768 ,,,,, ó 794!1- 410--- ---7911: 384· j~9~ 840 1é46 606 ,,,, 384 7946 i1b4 7996- ~ '1."19'- 48o ,.,, o- -,.,,. ••a 7t47 :a•o ,,,; &!6· 1798 480 7148 360 78-8 ''' 7948 168 7908 876· 7799 . __ o · .7:149 6~2-- -'lltt. . .. Jste . 7949 -- - i1~8: 7~~! o --,a--oo--- 33,, '*"o 11•o '•öo o 791o tooa 8oao 144·

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N.

Page 36: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

--- . -··--· ----·- ------------..,------------ -· -----------------·-------------'

NUM NUM

4 8 o __ ~-11_0 1~ _____ ttiO_!_ 8-lf~1:. ____ rL __________ ---~.a_4_Ö1~-------- __ .6112__ _ .8411. · __ . _____ --1-'B----- _ 825~ -- - f 8802 480 8~,~ SlO 8402 1.8 8412 191·

__________________________________________ .$2~.l ___ 1JU ____ ~---~to3 _____________ ~---o _____ ---------•~l:t: _ )!,___ ------'-~Ö_3___ _ __ 4Ja· ____ .!.413- ____________ .,.._,_ _________ _ 82!14 '"' eso4- o 8~,_. 1440 8454 24à 84!14- îlóo

· 1i;! 1~l ---·11~1 ,_~i: :1t:· ;-:t ::~: r::: · ::::· ,,:. ---------------------------------------------------------.8ll7~----···-- ________ _l_a_4_· _____ uu___ - __ 72.0 asi-1--------·'---------- ___ 84Ö.1 0--- ----«!J'tc.: ___________ 431-'-·---

8258 1102 1808 192 8J;e ''' 84§8 311' 8418 340 ---------------------------------------------------------------- ________ 82'9 ___ _ ____ __1_2n_~· _______ IJO.f_____ 9_6.0_ _ ___ .....c..S!I..t ________ JL_ __ 1409 _ . ------ - 6.1.4c ---~-llUO--------·-----l0-0&: .... _

8265 aas 811o ''' 8$60 176 B41o ?44 8460 24ó ....... a2.6.L. . %.2~J- lU~ o !U~-- . 1U.6.. 841.1: 160- -------"-'~ '44·

8262 624· 8812 4;o e$82 118 84i2 ó 8462 1200 ______ 8.2__6_! ________ _a:__ ________ ll1.3____ . __ !!t_6 _______ lllL _________ 844 JS4!-IJ·___ -----1.14. ------JJ4-43-- -- --- 0

8264 ''2: 8!14 744 8364 288 8414 i718 8464 110 ____________ a.u'~-------..J1..i _______ au .. s ____ _ ___ tu ________ _Q._.___ 484-- ...... .e4t!L- ----------------0~-----------•••t-- ._,, __

8266 Jl2: 8!16 o 8$66 1344 8416 144· 8466 i111· ... 8.2.6_,_______ 12tt____ JU-' __ .4ll1L. •a•-'~ o 8411 ieöo ,.,,. :s•é,

8268 102: 8118 06ó 8368 S20 8418 314 8468 4!1:

---~---------------:}~1- ~-~- ~-~·:l- --- ::;}------~---4~----------{11: -- ·t:::-----------::;:---------- ---:;:------------::;~ . '::;:-- . _ _ ______________________ t2Z~- _ _ _ _ 0_ _ _____ l_U_1. _______ .J.2..2.L _________ 8.$'1- ____ SIL __________ !4.U: _______ --- .. 86··- - 8471 - - -tJè- _____ _

&272 1~4· 1122 7&8 8!J2· 48ó 8422 :st2· &412 a~ó .. 8.213 . . .. UtS .. 8!23.. - 218 8~~3 _ 720 .8-4-t:t _ .ti 14-1J: 41ó· -8214 $64, 8824 720 .,,. 128 8424 648 8414 1241

_ -~175 . _____ l~~---- ______ .JJ..tf ______ ~----840 8!l5 ___________ __a ______________ .S4.H~- _________ /i.f!•-- _ _.,_,! _____ ---'--2-Ö------------------------------------ -:~;;_ -_______ ::;~-------------:::t ________ _!!~--- ____ :llt ____________ :~~: __________ ::~~ r::: ________ ::;J·_ 1,.,:. &218 552· asaa 288 8S;e •12 &428 2i6 8418 s;,. 82'19 . - ttl9. 060 . »;, . 14-0 . 8429 ••t __ ,._,, 0 8280 7~0 8Î30 1206 8$.0 0 8430 612' 848Ö 336· _82_8 __ 1.:.__ _____ _1_0.2~-------- ____ e IJ11· 4.32. __ .8S.I1-· --------1168.. -- 8431 --0- ---------8.4-81--- ''I 8282 120 &132 t6e ei•2 164 8432 • ~0 8482 4!1' e_2Jt3 _________ 38_4.:.. _______ ~~~~l _ _ o6o__ ___________ etu_____ _ o 8433- -t2tt-4----------~8.483 _ 7to 8284 o 8!34 1440 esà4 s2a 8434 864· 8484 480

.. ______________ e.a8.3.~ .1U4· ___ IJl!: - ---Uis 76-1 . 8435 ,,.. 8415 431' 8286 120· 8S36 384 e$.6 116 8436 41ó 8416 1214·

-------------~---M.87_ _ . .D ____ 8137- ... .61-a- 8-$8'1--- .10--4 ..... !-4-3'1 ··----lU·-· --14-1'1 0 Ba68 1•2: 8838 384 8*88 4S2 8438 !tl8· 8488 2t6

__ 816. . 9ttD __ l!3.t______ __61~----- 8:!19.. 108-0 .8-439.:.. _____ - --- -0 8489 1'111 8296 529 8!40 384 8$~0 864 8440 ~336 8406 ,, •. 8.291_ 1128 8141 480 . 830.1 0 84•H. t440 8401 410 8292 240 8!42 864 8302' 288 8442 ,,, 8402· 336 8293 5~2. ...8.!.~..Jc....... Q AS$:5 76J .. 8443 --264• 8403 1'16 829-4 '728, ..... 288 8304 816 8444 0 8404 6!11'

___ -~-~--~' _l _______ e_t~~---- . _ .124.8. 8!P!I~-------- ---- .2JJL_ - -- -.8-445 -· - --·- 16 0 849!5 - -0 8296 a•o 8846 t6o 8396 4!16 8446 -!1'16 8406 216

___ 8.2_2_'[ _ .964· __ 8!41 - 218 &j'i 763 . 844'1 0 8401 384· 8298 &o8o Bl48 o 8~98 384 8448 48 &498 ts.s•·

____ 829_9 ________ .31_4.:. ___________ 8149. ---124.8.. ---83lL------------O----- - . 8449 - ''' - 8.410-------- . -!5'16~ B30Ö Stl4· 8150 720 8400 240 84!10 l260 8!JÖÖ 288

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MUM N. NUM

8551 .. tsoo ____ ,f~i_'-----~------a _____ e6o1· 1-t.,a ..... &6!11____ 112 &.1öi-- ... 1&-&--------~ ---.-- ... _________ -~----------------~~---------~·-esoi _________ -48f 8t!52 !U 4 e6ö2 672 86!92 tta 8' Ó2 614

---- ----- ----- -----~----------- ---:~~ --- 12~ ----:i:!"------- --;i!---- ·- -- ::1: - !l!: ----- ----:::: ---------1:::· ---- ~;:r-~ -- ---4t----- --_e'a' 'a~$: tll.'- '-•-• a6iJ---· Ja, ____ . 86e -a ''ó' ta••· 85o6 649 .,,, 1•2 ••n• 111~ 8616 110 .,ó, 11sa: ~-~ö_, ________ 4_:1 2: ___ ~-----•nt ___ ~-------••..rl .. ___ .. eu7· _____________________ (t __________ -~'-!-'-- ______ i1 ö• .. _______ _..,.o_, ____________ .u.j ______ _ ~sö8 o ISD8 624 &65a sae 8618 ioös- e1oe ••ó -::~l- ------:~}t--------~l~~-- -- - ~-- ---- -::ïl------1::t --- :::; ::: - --~;n------------ ;::- -8511 0 .8961: .10'6 16-it: 33.6 8661 .... ,. .,11 ó 8512 48 1162 480 86t2 384 8662 118 8'12 396·

---IIH---------~Il---------tl~~~ til----- --~ff!------ · -::: -:::~- --,ö~-----------:~t:---- ---- 1:::, 8515 384 8165 !576 86{! D 866.5 -- .J!.l'--------•1-tS'-- 384 -----------eii6 _________ 6'2; - --.,,-.-_- ---- ··--·---aea·----------~li ~---- ···- •8 o ··--- --- ·86 66- ia a a e '16 161 ·

:~fi -:6r 1:-:: 480 :tt: :~: :::! , .. : :;t: :~: _________ Ht9_____ . _ó · ____ IIJ..i.. ____________ _s_~6. _____ ...... ..86&9 -614-- - &6j9--------i-4l'-· --- -----8'!1 9 ··· 0 ··· ··

8520 384, 8f70 816 86tÓ 288 8670 816 871Ö 384 85.21. . &112~. ..A.t.·'lt: ...... --~---8-6.4... . 86t4. 1344 - 1-6-11- -- D · 8111 11-81 8522 ld&ti ·~72 0 16i2 160 86'!2 288 8'112 ,., •.

. 85-23 lA~~ 8J'3- 888 A6f3 0 16·73. 4!2 8,13 671' 8524 312: 1974 624 8614 216 8614 710 1714 431· 8!25 _ . t6S~. . ___ !J.V_s______ o a-tiJ· -···- !J7-6--~~----36'1-!9-- ··-- «t 8.Pts-------------84ó--- --8526 ldQ9; .,,, 168 86tt ,,6 8616 710 1'!16 888

.. -8!27 --- -!L-----~-U!.z. -~------1152.. _ _.,,,: __ ----------t-0--4-· ______ .._,_,.,:_ . 160 .. -- &111-------··-·-··-- - ó· e!ae 1~4- e•7e 384 86i& 28& 86*8 {114 &718 264· 8!!129 . - .tU: . 8179 10 0 a . . . 16-tiJ - '144 16~9 · G- - ··111-9- tJSf 8!!135 432' s•so 384 &630 1296 86so 192' e?sö ?tö 8531 _____ l2.t1JL__ .8J81 144· . .8651: ____________ -0- 8611 -- --fl-8-----------1-11!-"--- -408 8!532 5,6 .,,, 168 86.2' 240 8682 4:11· 8732 d

_8!33._ .. ______ 5_1?_6 ___________ !t.l3 0 -'6-3-3-~----- _.,,. 8613- 3···----------~--31~3-.----- 671 8534 864· 1984 336 86~4 1856 1684 336 8734 911

___ .a.sn.. . .. ~--·· . ajl! . 11B2 ••îJ· . 384 a• es !f14· a's' o 8536 384· '''' 1214 86•• o 1686 1aó8 8?36 211 8!37 120 8i.87 .. MI-2.:. .. ----- .. A&i'f . '168 8687. ---·-- &jJ1 431' 8538 912: 8188 336 .,,8 384 8688 0 8138 *•• 8539. ... . .40$: ••-'-' __ ----11t!6.... .. 8ts•~-- o .. 8•••-------- --~!12&; 813o 1010

----~------ --e!~l- o e,•o 614 8e4o o 86oo ••o 814o 188 8!14! 72D 8lf1 0 86.1: 111 8601 624 874i :l.Stl: 8542 480 8192 144 8641 1108 8602 240 8142: 864

. 854:1 _ a s.JtJ_ ••o 86•3 384 _ __ ----8613 la a• &743 o 8544 240 8194- 1632 86.4 4:12 8604 i111 8744 504 8..J4!l_ 4.80 __ Sll! --- 480- 86~5· -134~~ --------1-605- 0 1745-• --1-'6' 8546 i7ae a''' 336 86•6 t6o 8606 76& &746 641 :::-: -~-:: ---~t:: ~::· :::: ,,-:"- :::: ;:r · :;:t =~:

... 8.!49 ..... ------S-3.44-". __ . ______ _a.Q.g__ ·-··· 0- ···-36+'--'-·· -----~~Q.t-Q------- ---8-609·· 8-40-- ······~---&,_.., ____ ······ ·····!1'6·· 85!55 !!J!t.2: 8100 600 8610 600 8700 0 87!15 liDO

......... ·---'---·-~-~---------..................... ___ ......... -------- ---- ------ -·----··· ................. "-----------------~ .. ------- ... --

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'i U~ . .tL ........... -·· J~Ul'L N~ -N.UM --N-- -NUK N NUM'

0 __ $J11.:_.~-~----8-~4..:~------ ..... 18f1' . 5IIL ________ ~8.Û1.:__ _______ {il6'- 8911'- -- ··------0------8752 72: 8A02 11'2 88f2' 104 8952 s•s· ltlt· 3~6· 8 7 5 3 _ _ __ ~J~-~~ __ _ ____ t!O~-------------- 216. _ _ __ _ llf 3. . 5.7.6:.. ... ------~.89.~3____ __ ___ 11 ..... - Bal !L . . . --~~a.Q -

···--·-----··--·-·-------------~--------i?5'4' 864· 8804· 768 8$84 960 8904 410 8914 13~0

.tl'.5 4~0 __ .li..05.:. _ _ ____ ..t.6_o elp· _ a . t9.ês 5'6 8tts· ,;, 8756 7~0 8806 672 8886 ,,, 8906 !632 8986 ó

_______ -------~ Z~!'---··· _ _ __ 7~_o._ __ ~_JJJ)_7 ________ .. _ ....... Jt.... ___ 'ë't~------ _Jf_o _________ eJ.fi7 !SI_ eL. ______ at.!i ___ ---· . t.s6 ... . 8758 480• 8808 240 8818 1808 89Ö8 144· 8018 110 8759 ___ ---~--------$1_01___ 168 _ 1li9.: _________ _1_14.:. __ 89Ö9 itiO ___ Jtl8.t__ ... Ó

-- -8,-60___ 288 8810 1248 8À60 0 8910 Ïl96· 8960 48 . 1.161 ... 6a_4.·. BJU.i~ 960 .. 81.61. 16.!1.6 ... lti1 a..- ao•t · 1S.II 8762 13~4· 88l2 120 8862 672 89~2 216· 8962 384·

. ____ 8Z_63_ ________ l~l'----- _IUI.:S_ _816. ________ 8.JA.L_________ _Q __ 8913 Ul·------_1163-- 616-8764 o, sas.4 ''a ata4 264 89{4 718- 8t64 &76

________________________________________ .8J_65 ____________ !12~ _l_ll_l _____tL _____ ..!&6L_______ 1j4Q.__ _. 8ti5 _______ _,JU'---------8.96-5 t41i. 8766 1296· 8816 0 ••• , 612 89i6 .• ,6 8966 ,.,.

-~j:-r 9~ :tt~ - · ~- :-J:---- :~:~- ~~: ::i~ t::t :::: ,.,:. ________________ e7_6~----·----t!l,l .. · _&J1i __________ 11!6 _______ .s_.,, ---••n ........ tt.ii _________________ l) .. ' ''' ïonaè ___ _

8710 §18, 8110 504 •• ,, 1001 8920 240 8916 960 __ e_7_11.·.. .. toe~. __ .t_ta1. _______ !UI.2__ __siJs.... ___ _n ___ e.t.tt--~·-· -----i-'-'8'-- .. 8911- 486- -· --~- _

8771 288 8821 768 ae;a 312 8912 480 8912 360 8773 4Sri B8i3 o a•ia: Jta.. __ aoa3-. -- - _.,, ao13 . -&40'-· 8774 2016 8824- 384 ··~-- 1184 8914 0 8974 768

····-····-----···-----~-------- ···-----·-·--------------- ... 8.175. ·- ___ ft .... ____ ..t!ll.:. ___ ~--1!•-•~-. -- ..•• ;_,. --- !17-6.--------••.u----- --6'-1' --- l91J------------i----e??6 !84· eaa6 too8 ee?~- ''' 8916 576 eo1~ 10~

------------------- -- 8'111 . - - .lQ.I5_!.. _______ !!J7 ---- ------2!8-- •• ,_, - ---···-·---'8.0.---- _____ J9.t.l-7-- -- . -0- -· ,_,, --48 0 c

87?8 5~6: 8128 0 •• ,. 384 8918 218 8978 804 .,,, . !6a •••• 12a4 u-t• . __ -"o. . »~• ,.o . .,,9 ,,,_ 8785 384- B&lo 816 8110 96 8930 i0'6 8986 184·

···--· _87.81 ______ ...i.6Jl _______ lt81: .. o. --!IÎ1·---------nG--- --- -·- .. 8931 --11-8-- --------U·-1~---------. 1440-8782 336 81!2 96 88Á2' 912 8932 192' 8982· 9!6

_________ 8.1.8.3 _______________ _à~--------~8J_3_ .. . .. ____ !IS _________ IJU~-------'Ut8 - 8033. --- ---A4Ö---------8.t8-J- .. 0 8784 360 1834 1112 88è4 216 8934 960 8984 936

__ eJa!L ___ -----~'-tt __________ . .!!J!_ - 3&4.'-~-------t-ils at-6- •••s" .. o· ataJ· ''' 8786 1245 8&36 294 éè$6 '20 89~6 408 8986 984·

·-----···--·--- _____________________ 8_1...8'!'_ __ 2.8 .. 5- ... 8U'f__ _ ___ 9J..6 ___________ 8-Ü-1 -0- ·--·· -89-37- ------J!Il----- ··-- ·--191'-- -· ,_,, 8188 336· Bl38 60o e$ia 624 8938 614 atae ó

_____________ 8_1..8.9~ ... . ... 10.~6.·_ _tt.li _______________ Q______ - eti9 1200 -8-9311\-------- - ... •.til· 8989 12Ó 819Ö 168 8!40 612 8890 .,, 8940 218 899Ö 864 &191. ó et41 - -1tta: ltos.· '168- 89-41 - ,._,, 8tt1 ê 8192 432: 8842 312 8801 0 8942 itll 89~2 06 8193 720. ___ 8.843.------ 6.14 -8-801. . --5!12------1-94-3~----- -- · ·D 89t:5 Îl04· 8?94 5~4· 8844 384 8894 1tS2 8944 Ö 8994 1ÓS6

--------·-------------------------··----- 8.1 9.!1 ____ l.l!L .... - ___ J.J.U--~-----------384. - 88 Oe 8--- ------8949 ---Ï632 ··· 8-9-t!JL------ --UQ~- ---8?96 0 8.46 1680 8806 12 8946 960 8906 480

.8.1.9.1 114: -·-----&t41 ~ 8-&91 .,_61- 8947 ·248' 890'1 ,, •. 8198 912: 1148 96 8898 480 8948 360 8998 431 8?9_9 ___________ Q_l_ _________ uu.-1S92-- ·-----u••----------3-1.6----· &949-- t•o - ___ ,,,_, __ -·· -o 88ÖÖ UtS Bê!ltl 960 89ÖO• 120 8910 !JÖ4- OOÖÖ 110

Page 39: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

N "U!.il -·----- -·----··~--~·----------------~-~ ---·-·---~r_, ___ ,, __ -- •-··• ~ •• N NUM "! .... NUM

Page 40: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

NUM NU!L NUM

-~~~·-----·---~-·-·---·-··---~-·----~-··--·--~--~:i;~ ~;: ------:}~i-. ----~~1-!----- · --:l;i lal-- -~--!:-ii~------I-~:-~- - • :::~ --· ::!~---- -9253 .... l~-~;----------~IJlL~-----JL _____ ..... ~ifl... _ ---'12 ... ----'-~-R!~------ ··---~·H4: __ 94f.3. ---"~~~---- --··---

-~--------------- ''929-.- -. 816 0104 264 0314 612 0404 0 9414 614-

-~i-~~ 33l ·ll-~~ . tg:: :tt~· ~=: ::~: ·::: ::::; . ~ ---·---·----·-----------------------~l~l-- ~;;~T------ll~l-- 3t·~ --:=z~ ------t::--- --::-~~ .• :. --:-::!------ ::::-------·-----·----··-----------·---·--·----·--- ------------'--2'9 __________ ~!!_$ _______ f!O! ____ . __ J._OI6 93J9 ____________ ft____ ... 94Ö9 _ ... !12: _______ .....9.41J ___ -- . -.600 -

9265 432: 0110 432 os•o 2ee 04îo et& 046Ö 384 ... 9.2..6.1 . --'~~l.... . 9J11.· 0 9.ll1. 6~?2 . Otlt1 110. ..9A6i:. . tSI-1

9262 336- 9112 102 9$Á2 $64 9412 1tt 0462 ,,. ________ ~.2.6_3_--·------------o-~------'.t13.. te .. _____ t.ti.l__ ---••ó-. _ __ o.413... . _ .. r•••~---------•-•u.-- o

9264 192' 9S14 1344 9,~. 360 94!4 {111 9464 624 ···-·----·-···-------9..2.6.S _______ i1_L __ -_____ 931.5. .8J.1l ____ O_JlJ' ___ - -1~04--- -······ --L115-~--- ---------·---~--------7--046.J:.---·--·. -·l'1j - .

9266 1128 9~16 314 OSi6 960 04!6 672 0466 416

-·-:~ti ilt :tti ~~t--· llt:· so1 ::r: ·:11 :::: ··: ... -·-·----···----- --::~t---···-----1 ~:-::-·· -----lli:---------,~--- ---- :tJ;- .. ····-······ 1 ~~=--- ------:-:;:-- --------. ::-: ····· .. ::;~ 1::;---·

-----------------•-2.7-1 . J) _____ ltli.:.. _______ ,_,.a ___ - 93,1·. 1~1.6--- -- 94.f1: __ -· -'102 .. 0471 --0--·--9272 4!2· 9822 528 ·~t·· 0 9412 1112 •• ,2 24 9!1! 76li Jl!l- . 696 9S-1J: 384 .0.413 0. 04'3 9-" 0274 18!: 0324 410 ,,,. 1206 0414 101 •• ,. i111·

____ . -·····- ______________ ---·· ...... 9275. - - 9A~L ------.lU!.---·-----840 -·- t~lJ ----0-----------0...f!J,_ --ÏI44· --- -.041S·-·- -· - -:16!---··-9276 0 0116 816 0376 216 9416 i111 •• ,, 864

. ---·------·--·'2-'-1--- -·· -·--ll~:--------'121-- ---- --a --- oá-!1- ---------'-'·••------ . ._. .. ,..... -· IS-6- - ,._";- -· - --- tt't'···· 0211 ~6o oeas o os'8 12j 9418 sö• 9478 624· 027-0 .Q.e ......... o_p..o. 1584 ot~· -!J1-6· ktJ-- ~~·· ... ,,,__ . 0

. ___ :l:t_. ____ .!til:_____ . :tlt ::~ u :Jlt .............. !;t__ :::~ -- -~~t-------- :::lé .. -u 1~::. 0282 ,,,, 9132 408 93.2 552 0432 360 9482 864

__ H83 _______ -___ 2.t,4~----······'-l33-. 120--- .. .9,83------ --- Ó- 04.33..... . .... 3.84-----.0413·· :184-9284 72~· 9834 1008 ·~·· 288 0434 i016 9484 3!t

......... o.aa.s... zt...s._ tt3.! osàs· &64 043!J ' '' ,,.,, 'Ht&· 0286 ~3&- 0836 518 ·~·· 1112 0436 . 0 0486 960

_______________ ...92.8.1:._-- ........ Oe .. _ -- 1537 -----·- 6!4---- ----9!3-'1---···-- .... --fl-76 ··· 043'1---- --tf-1.0- ··-9481··--- O· 9288 384· 9S3B 960 Oii8 !20 9438 614 0488 288

_ -- ____________ f.JBSL .... 48Ó . - 913..9---------410 -..OÎAO -- 1184 --- 04:19-----~ 0 9410 !008 · 929Ö ~2~9: 9140 0 0300 .,. 0440 -~0 9490 ,,8 •2•s.- ttts, '''t .. ~'"'" ,_,",. o 9441 · r""••· . t4f1 toso-9292 192' •s•a 576 o3~2 ••• 0442 410 9492 aaa-

---·----··----··----------'293 6tU _______ .tJ.J.!.~----------o. ........... os•s -7-U------9-4-43--·--·- - ••o- 04tts 431 9294 12~0 •••• 192 .... 4 768 9444 431' 0494 1144

··---···--·······-·····--···----·-----··----····---·---------'2.9! .... n.:. .. __ U-"----·-- 768 9SO!J'_ s_,, __ ------'445-- .. ,,. ••••-. - ---ó ---9296 336t 1846 720 ,,,, 648 0446 {7ó4 9406 264

.. 9..2.9..1.. .. 120. ------ .J.i''- ... 576 ,,,, Jf)4 ••• , 0 ••• ., ll64· 0298 432: 9848 314 .Oftl 864 9448 3ll 9498 ,,,_ .92.9.9 ________ 1._tá_Q ________ 9J.4l_-_______ -6~UL 9-S.CtO ...... -----.Ó-·-···- ---0-449 {302· ---0-4.9----· -$11 9Soö 3:u. 9slo 168o ••öa ••o 9410 is••· otóii o

Page 41: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

... NUM N

95Ö1 §'/?2: ~f~-1·-~--··~-~-~0... 96§1' _,.,, ~96.!1...... . .&.71' ''Ql - .!8!.~L ---·--------- ---·--- .... -- ..... -............. - ... --.. -·~---9s(f2- -5'16· 9812 144 9602 10!96 96!12 288 9702 !ÓSO

··--~ .... ---------.. --------~·--·------~-~-------- ~{i!-··· se·~~ -~-::!--------ij-f~ ::f: · :::- ---~!:f·· ·· ·- .. f~;~ -:;::~- ----6·4:-~-- ... O_!ij5. 5.~J~. Of'-'' 6?2 96.g!5. .1Î!l6. - 9-6J5 0 · 07:05 · 864· 9506 1680 ·~-6 408 96Q6 816 ,,,, 864 9706 ,,,

_95J)'1 ·-- ___ ...... _l~!·----·-----l~J_l____ __ 614.. ___ .tJo1_:_ _____ -------9------------·- .96!7. .a••--- . -------''l)-'-·-~--- --•••--

-___ ::~-=---------'1:.f .. ____ ..,;t:: 122ri -:tlt_ ... --~-ti~t-----·--·-=·:::_- .:.::····-----·~---!;jt~------1::.

05tö 7&8 '''o 5'2'6 t6io ?'2 9660 a t7iä ióoa 951! 0. _ 9161.: UU18 . _ 96L!: 18!6 1661 4!16- ''is.· 0 0512 120 •• ,2 384 ••ï• 202 9662 768 9712 0 t5J,..3 _____ -----'-'.tL ___ -___ __.,j3_~ --- __ 432 _________ ,_qJ;____ ..... 1 o• ____ ' ' '3 -- ........ --o --... -~---''~3--·- •a•--0514 624· 9164 0 96!4· 1!536 9664 0 97i4 160

... __ ... _ ........ 9.!.1$ .. ____ ~~-.l'l.L~--- --- ____ 9J6.! .. _ ... -- --·· 91~ ... --- .. ·---'M''-.. ---.. -- -:-·-.... -~ 0:46&- -- -----itu+ .. --- -971_,... 5-'16· 95i6 384· .,,, 11!2 9&!6 240 ,,,, !536· 0716 671

::t~- t~~t- :{:: ,~- - ---·:-!l!·· -1 :~: :::: -::::. :;i~ :~: ------·--.. ----- .. ~------~!!..1~ __ _a .......... 9Ji.i_~-~ .. --1H4 ____ . .$1419 -- .dl.- l-649-·--------~ .. --741" · 0'14.9 - t--·---

0526 os tl7o ''' ••tó . •ao ••jo sas t'11Ö 61•· ___ .. __ .. _-f.!2! .2.5~ . .6- ------'J21. ___ ~_ .. __ ~_81L.. ___ t6..t1~. -- .. 18..0-t-- --------'i-6,-1 .. -- ---·-·· - -0 0'111· 111J~-·-· -

0522 9do: '''2 71o ••ta •sa t672 211 ''ta· 1osa· 9523 a.e.s. 9t73 41o oals o ... ... ''-'--l s'' orts ,,,. 9524 4~6 ,,,. 648 ••i• 220 0674 1302 0,.. ó

" __ .... ~' 2..5... _. . ... _l_2llJI. _____ ·--~--•..~.t,.:. .. _ .. ___ --- --- o_ -- ... 0.6 IJ . . ... !.'1-''-· -------- .. -147!.. - - IA 0 · ''*'' -- -... 4.-JIÖ---- · 0!26 5~6 0176 720 oai6 1Ï8à 9676 !02' ''*' 1206

·--·· 05 2'1 .... -- ------ .. -...ti·'~-----...... 9.117____ _ __ ... 314 ·- .... 94t-' .. ---- .. ~-·~-.. --- - __ ,,,_",_ ---il,.; --.. -· -- .,_."_~_ -----------! .. ····-·· 0528 336· ,,,. 1032 t6ê8 0 0678 ~32· 0718 71 9! 2~ _. · .6.c'12: ... P.IIi . '1 68 .. . 0-4-tt . ---1-Ut------ .... uu.. --~1Ji- .. -- .:.O-Jt9-'"- -- 9~& 0530 12DQ OliO !92 t4~ö 120~ 0680 .3!2' 073Ö 480

.. 053L._ ...... __ 5}ll_ . - _____ fl-81· 1.S4A --- ... J-6St: ........ -.. --------·-O- ......... __ . 0611·····-- ---1 .. 016-·---------0-'J!-c------·-··· 168: 0532 Ó 0!J82 720 05~2 218 9682 876 9732 I~Ó

.. -~5_3 __ 3 ___ .. __ -~_l~t_6§ .. '- _tie 3 __ -t _ __ .. ~~--.. - - ...... lto- 96-13 - 4-1-t"--.. - .... --.. O.,:s:J--.. - - so 4-0534 96Ó 0Ï84 0 9JS4 672 0684 ,., 9734 1344

..... !.53.!L.. .. .. .ó. . .. JU! .16.8 0.6-fl-- 111-6 96&1 . ;410 0715· Ó' 0536 16! .,.6 816 .,,, 480 0686 1016 0736 384 _O_!.:n .. 916· 9!Jl1_ .. ___ J.!!-2.:... __________ 0-431- 624- . -- 0.617- . ~-- - 0--- ··· 013-; ÎOS6 9538 384· 0188 384 ,,.. 012 96&8 _2~0 0138 , ••.

------------9530. .. 112o ota.t_ t6o .:o•s•- o-- -•••~~------!14&; ''3' . :nt· -- 0545 480 0190 1-40 ~ii~ 202 9600 768 97•5 IQ8

.9541. .. !16.4· 98 .. 11·-- ·- 0 06.4t· 1ll2 ,,., . 410 . 9'14t· 96Ó 9542 6'2: 0192 24Q 06f2' 6'2 9692 • 0 9142 528-95 43.. _Q.:_____ li.R.3--~--.. ---9liO J).-4-3:. . ... --264-·-.. ·---.. ·---96--03------.. -1-tl!c-·---·-· · ·- -0' 43 -~- ... ----· ..... ~ ......... .. o544 384- ,,,. 1200 96•• ss2 9604 os.a· ,,., 1oa 954! . . 12J48~ -.. - .. 9!i.!.:.-....... 314 0645' '-···-·--·---'6--0.J ·· ... -- 0 -- · .,.e-L .. -- -· ---J-1?6----9546 lz-s, oeo6 6 06•6 612 0606 t44 o'•~ t44ó 9.547 . U2~ ....... 919 .. 1 176 0641-· -- 0 9697 .. 41..0 914:1: 6~1· 0548 289- ,,,8 480 0648 t20 9608 Ï301 9148 216 9549_ _ ... i! .. s-' ... -·-·--'''' _ _ o ... ~ ••••---- .... _.,,a~ '"''~ - ,,. __ ---~-•,••----- !.-944· os5ö 672: 910o 110 t6io 1200 o'öo 336 97Sö ttll'

Page 42: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

~ ... NlJ;~L- .... N..... ..N_u.M ...... ~N MUM. . . .l'L. . ...... MUM" .N N:Ufiit:

-··----·--- ·-·~·· ····--·-··-·····-···- -····---·-·-··· -·-····-·--···-··-·-=-9_ =':..~";'---·····-·········-···· ... _g _ _ _______ OJ_Q_~-- ____ _ .t~l ~ .... -..... __ _ OJÎ1.: . . _ _ _ .lllD _______ l9 61 ... --~ __ l4. 4 __ . ~-- _ 9_911~ _ .. .. . .... _ 0 -9 oaoa 3to oeie o oofia reoo 9912 t••·

---·-----~t~ 529 9103 888 . __ fi!L __ 4.32 ________ lt.Ö3... ······--7-D--~---··9.9.13~- .... - -.10!16 .......... . --------~--------~-------- 9154 ........ ····-g-~,---·---- •• 0.---------iti ,,,.. 1 ,.,, 99 64 120. 9984 i 'lil'

........ 9. .. !.5.5 4~.a. .. . .. ~a.os. .6.7.2 '''-'~ o tJ.öa i:O-J6 .;,,,, s••· 9156 0 9806 1680 9.,, 96 9906 6'12 9916 511

-----------~--------···---·-···------····----······--· __________________ !!~.'! . _ .. 6~~~-- ---~JOZ 9 _01.1.1·___ __120___ 99.Ö1 . 31Jl ... __ 9.91,_ 911~ 9158 . 16! 9108 120 •• ,. 6'12 0908 416 0918 431' 91!59 ___ _O·~-------..!l0-.! ___________ 1_,11~------- ... tllt·.. 104 ~.990.9 Î144· .Otlt.- .0 - ·--------- -----i-,-6-o ·· 1~4 ou.o 11o •e6o 314 ooto ,,4 ,,,ö 384-

.. __ 9-'l.6L ~-~!2.: . . tll1:. .5.D.4 UU: 11!.2 9.111: 0 tUi '''· 9162 516· 9112 624 •••• 104 99i2 314 0962' tÖ!I6

-------------···--·····--~··--···~- . .... ______ ............. 9'l.63 ___ ··------ . .J.~_j'----~-'-ll .. ~ -····------ .. _!.te __________ ..f•~~------------ .............. .tt ................. to;a................ . ...... ••t------- ... ..JJ1.63---·· ---·······-···· 6~4c- .. . 9764 Y12' 9114 168 9.64· 220 9914 93.6 99.64 288

----------~------------------~-----~---- .. J!.16.!... ______ ~UU: 9 IS. !I· _________ Q. _______ ~-UfJ~' ---~--33.4c------tti1-------·-··-··I-'-6-------~-0-0U-------------ll6"----···-·-----····----·····-- ........ . 9166 1ao 911.6 384- té.66 1&48 ott6 o 99.66 1248

..... -------~-?.ri .................... ~--------------!W- .................. ~.J:A. ....... .t.áit .. _ ... :UA ... tl!.7. . ..... {10.4c . ..tul·---·· --~ 9168 J84· oat8 11?6 •••• t68 9tïs ilöo· 9968 ó

·--·····~---~···-· -------··---~---------ill9 _____________ 6_! 4 .. _______ f....l1.2.:. ... ----.t.t2..•--· ····· __ ,_.._,_ ·--·-- . .il-68..-~------ .Qtt9 ............................. o:...... .. ...... J).0-69---·-····· . ·····710 ............. .... . ........ . 9?76 lOÖ9; 9620 0 tè1Ö . 1?6 0910 96. 991Ö 118

_ t?.'ti_ -----~'-tl'~-----~•-•a..t------------''D--- .. ·--•.J.tt.:.--------------O------ .Jlttt.... -- ?ló. .. oo11-- -1 oa-ö----····· 9112 144· 0122 120 98'12 ~·· 9912 614· 9012 3&Ó

:i~:- ~~i~: . ::·:: 40g- ::1:: 1~;:· :::! !~:. . ::~! .:::: '''' . ----4-- ...... -•us. ---10.4t .. o.f.tt' -1t!la . 091s ... -Jö4- - _,,,,;_ - · o 9716 240 0826· 864 9816 614 9016 Ïl44 ,,,, 140

____ 9'111 .72.n --·· ~'-817 .. 6!2 ,._;.,:_ . u.o .... 002-1 0 ,,,, 911'· 9118 ,... 9128 384· .~,. 6?2 9918 288 9918 12ó ''-'-2 . 11~l2~ _ .. ,... . 104· .. ?. 0 9919-· tl14· ,,.,, 480 9780 184· 9830 1200 9880 384 9030 1.68 9980 0

... t1.8.1. 6d.O __ ·- ... 9&31~ 0 . 9.Jlt 13 4.4 . - .. 9 931 5-lt · .. 9 tli tl 0 Ó ·· 9781 8.64· 9~32 216 9882 1080 9932 48ó 9982 314·

·······-----·- ·-----------·-··---···- ................... 9.l'Jt3.___ ~------.ri_------- ___ .!~3.3.. ___ ---11411 .. - ___ táJ.3, .... --- ........ AU. ..09:1.3 ........ -- ....... !J,.l . . - 0913 ·0 9?84 33&· 9134 960 ••• 4 0 90:14 411' 9984 0

_____ .... ~1 . .8..!.. . ....... l.4.4 !L ................ .9AU._ ............. 38.'--·-·········~··-······tJàS.· .672. -99J!IJ Ó· 9915 768-9786 lóJ6 ·~~6 456 9... 612 9936 314· 9986 158 97 81 ....... 2.~_4_:_ -·----~lil ____ -~-6Jl.Q _________ IJ.l7~----·---·--.a-------'''-'-~---- ...... 3:1.6 .. ~.- ...... 9.9.8.'1~-------· ---61-4·

---·----------··----·····--·-~-------··~--- ·--------------.188 à 98:!8 61~ ••• 8. !44 9938 ,,ó 9988 . 336

.......... ._ ........................ - ... ............... .91 6.9 ................ .J.Jjl_i. __ ~---~-1.8.3.9 ... -·--···-----.0----- ...... ltl.l.c ... _. ___ . ___ .. 8.6.4 ----- ---- OU 0~-- .. --· 6'1-----···-····· -09&9--- . -1--1-0.6···· '''o ''' t84o o oáöo 1632 oo4o 21e oo9o ''I 979t Q ll-'1 tOI6 98.1 'IQ 994!. tftó8 990!.· Ó · 9792 1~4- 0142 1148 0801 240 9942 410 9992 480

.9.193 .. .'-ID ... ___ 9j4J. ___ ... 18.8. -9&03-- - ll6- 9943- Ó-· -90-t-3- 614· 9194 1249· 9144 432 9894 10!16 9944 671' 0914 i1D4·

.t19.!s. .. 48!t:... --'1-'!' ...... 1!136 .. oaot'- o- .. 994!1 ·t•o-- --9t9!J t&ö 9196 480 9146 936 9816 936 9946 Ï111 9906 431' ~.,., 768 .... , 0 ••• ,.. 116 9-947 111· ,,,, 431 9198 o&o oa48 314 '''' so4 ,,,. o •••e 114 .......................... tito ............ - tt ________ .. f.-'9 ................ 1,.008 .. 0809 -'110 - 9949 -6ÖO 9999-· ó oaö5 ''~ •a!lo 6oo ooöo a•o 9o!lo i&ao 1oooa t!Só

Page 43: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

~ .... N.

11 .11t.1 .. .. ........ J.ll.. 4337 1 !11• .. iló.t. ------~--2.0-1-- ---1 t 98t· -2 i9· \'U 1.!'15 iöz· 4i18!J 112 ,881 2Ó2 1iÓ!J3:

___ -----------------·----------------------------------------- __ __3 ----~- ____ .. _?_-______ ,_3 .16.42. _ _ ____ fll"-- ____ .... 4:ll.!.. U-3 .. -- ----UU·--------------1-Ó-3-:.. ____ -l-ll40t 4 ll 54 1.743 l-§4 4481 114 .il!· IÓ4 itiO! ! ~U .. 1.1 . 174.3 0!. .4J53. 1.l5 t2t1 20-!J 1-IOJ: 6 81: 56 1791: 1~6 4615 i16 .211· lOt 11533: 'l___ ..... JU.: ____________ IL____ _ 1.839_ 1~'~---- ---~-"-''~-------11'1 -- 1210 -20-1--------l-tiiS: · e 9& 'a 1863 ~~· ''~' 118 lsas aö8 1ts11

.. .12~: ______________ _l2 ______ 1J_3.!L __ .. _ .. tU ........ - 4A0~-~--------1-1--9- tS.aJ -------2-ó-9 ---- --lf1-J•1.: -tt7 6o tt3s tio '''' i6o l4ot ai5 tt8os

11 . 1.71: 6.1. . ..... 2.Qlll... lli ••• , -. 1'1 J6tt 2i1 11,68' 12 1'9' 62 2103 1!2 . 4945 162 8701• 212' 13031'

___________________ 1.3 _______________ a.d_:s: _____________ 6.L .. ~--'1D.3 ....... _ ..... !is...... .S.e•l--- ---- .. -1-43----------------- -- _.,_,~:. 21!- . ti!3J; ... ---14 211: 64 as.o 9 }l .. • 5t37 i 64 laat 214 1110!1'

_____________________________ .. _____ 15 ______________ .2_!'-1_:_ _ __ .65 ___________ U_OS ___________ • 5: _ -· ... 5185 . --·-----1-65--------- --1-tllc- ---- -- 211· tDO-I-1

16 as7 66 2101 11' 51!17 166 ijo4t· 116 i3Sot· . 17. . !til! . --'' . U!!i ... - it-'' . 5J'' !61 "09-41' 1!'1 113tf'

18 l4t· '' as73 ti• S44o 1&8 •o•s 2îe 1,,,.,. ~~------U---------l..6.JL________ --'-0------~-t--69.----~------tlf: .... ~. ··· · ··· 5-44~---- ··· ·-169--- · · ····t-t1t-•---- - !1 9 · 1361 J; ·

2Ö 389 70 25!7 ttO· 5497 170 ÖSl!' IIÖ t36t3: ----------~ 2-1---- -----43.2 ~- .71 . 1511--------i-i-t-----·- ,..,., - 1?1· .j4Q·----- 21{· . 1180!1 1

22 461: 72 25!13 112 ,,., i 12 ,,,, 212' 13840' 23 -- 4!1-: 73- 2601'. i-iJ: 5-743 1"3 ''I!'' ~tt3 1~·-·i 24 481: 74 2,21' tt4· ,, ... 3 114 0171. 214- 13907'

.... .2!. ............. .!U:.. _____ - .1~L -----· ....... 1177. ..... -- --tfJ-~-·----584-1~~ · ···· 1'1!1. t1'1t·--·· -----·-2-H·~- ·-- · t·414,'·· 26 se1: 76 2801 it• 6e3s 176 ~''' at& l414S'

.2'1: .................. _6-U' .............. ~----'t"l ............ -2807 - -- tt-7~------- -6»t-···- · 1-7'1-- · t!-4!.-- -·-----217'------- 1416-1-28 619 78 2945 ,.. 6043 178 .,,,. 218 14411: 29 . lt .. t.; '' ., .. , . ~--'-'- -~~--- . 1;, tiG-99· ···---····t4t!t· 35 739 80 2969 ,10 6235 110 t51Si· 235 t471t· 31. ..1!.9' ····-··· -~~~- --'-1~-- --!0-11-~~- lil'···...... . .. -6P-5 ---- ----- ·111-'- . - --1Ö211' -. - -·2·3 !;~- ------t•11t~-- -32 711: 82 !!19 j;~2 6t&ÓS 112 1Ö301' 2SÎ 14'191'

- 33.~ '1:99• ---13----------J.l-11-- ··· tl3 · -6t&!JS;' ---·-·· -!-1-3-··-- · têlJt-J• · S!JJ ····--t·4fl9r····--··· 34 847' 84 !239 tS4 66!9 184 1Ö443 234 1S1!JSl

--~-·-- .3!. 8.91! U Q81.. tfJ' 66t0 - ti!J· tö•tl' ISt· 11203 · 3& 91J' 86 !407 !36 6667 116 10''' 2:16 11111'

___ ;s_,_ 9tu_, 81 _ .... no-'------------1~1:. 6'6a-. -117- -tj-811·. as1 t84t9 38 lOlt· SS !.31 tSI 6819 188 10811 238 111!1'

.. 3.9 _________ &.021:... . ... 8.9.... .U!!.------ itt9--- - 69:Jt ft9-- - ----!.it-tf 239 t15!8 4Ö 1o4S' oo saos {i&o 6979 100 1!o67 a•ö 115tl 41 ll-~1; . 11 !-'43~ i4!> 7Ö,_!J 191 1t06'' 141· tf6S9 · · 4t tt$~ 9a !743 i•a ?tas 102 tio~s· 242 1'''1·

.4..3 _______ _t_a.t.!~.. . 93' !791 -·-----i.4.J.. .......... ,,.3 . ····103-- " ·tt-tiJ-;-----~ ·-2•3~-- 118911

44 12~7 94 3811 i~· 7tl3 104 11343 244 1106, _ _ 4!5._ 13.11.1'. __________ 93 . 3887 ----141---------724-9 i OS · ti•tt-, ~ - ------2~· 16tSI·

46 1357' 96 !911 i•' 744t 106 1iSis· a•6 t&a1o· 47 13-5-!'. ,,_ 391J9 \~--7 ., ... ~ 10'1 ti6J-3, . .,.,. 162,9· 48 t:nsl 98 4067 148 7Jit 108 tt7ts 248 t637t·

.. 4.9- ......... 14.~9- ---·--;:-lt..---- 4139 itf·····. . -'1631)----- ---t-!9 1Ï'''~ ... ···t••-----···· 1611 ,. 55 &JOl: tDO 4169 t!JO 7809 200 11831 21Ö 16663'

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A!~.: .. ___ tt ... . A! ···~··

Page 45: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

--il~------~~~~;-r -11r -----·:-:~~:---------!ti~-- :i:g:- ::; .::::;;---------ili·---- ~:z;:: _·SÖ3 . 472!J1. ___ Î_,_~: ''5~1.. __ 6l:L _____ 62113 613 ... 6UA3; _________ ,.-Qj_,__ ___ ---'ll.Q!It- ....

-------------~------~--------- - --------5o4 -----.-7-4-oi ts• ''''' 654 •eaîs 6!!14 7ö~u.•· 104 ?aeao· 50!5 4~4-~7 j'' 548.91 6..Ö!. 62S2.!L AI.!J 73310 105 . 18Jt1· SÖ6 41833: f56 !54P63 6Ö6 62660 616 764{!!1· 106 18801'

-- --·------------------------- ------------------ ---------------------- i~-~--~------ -:i:: r------------t:-i------- --;:~-!~-- -:-~-:~----- ------!:;::~--------------------:::-- -- ----; ~-~·:::------------ ----·----·;t:-------~-;t::;. _________ _____________________________________________ _ ... _s ö t 4Sa~1_. ________ _JD _____ !lh6!. -- - . --- -6!-t·. . ---ih-S3~------&~9 --- -·-7io-! ~; -- --- -- 1Ö9'-- -----~7-tit-1-' -

siö 48489 ''o 55515 & __ :ó. 638?? 660 7it!t· 11_-ö 10101 ' 48 !'i· ' . * ... '' ' 511 .. 48". . 161 . .!J.2D.I •-u.· . 63J1' 66! 7t-JI?' 111· - - 7tlot: SiR 48501: 962 55103 6i2 13461 662 1{501 ,{2 '9597'

. ------~-----------5i3-. --- .4JUP.!-.. -- -- ___ jj"!, _______ s.&.o.U.:.___ 111 ........ 63J4t-:-. -------6U------1!1tt'-- '1l3 - ----- é70315"---514 48885' '64 561L5 6!4 63.10 664 1i1it· '114 Bd113

..... ,,, -------------------------.!1! ..... _ .. ~o.Ji --- ------Jl!I------5-UJ-9_________ --li-l-' ' ___ _.Jt49 . -6-65----- . -1i9tf:. ·'?l5'·~- . 8Ö •••... -516 49-113 f66 566l9 •i~ 64115 666 .,.389· 11~ 8~38, 5-17 &ilt-7· . Ul ------•-••u--------- ~~,- -t-4229 667 ,,,.,,, 111 eottt· s1e 49!ö9 t6e 56667 6!8 64373 668 ,,.,,, '18 8q111 1

_ -----------..S1.l-------ID.4t____ -'-''- ___J7!l!1-• ---------1-i'-'------ ----- 6440:5-c --- -669-- -- --···7t!Jt-o·------~- ~ -'1-19· -- -- eo 111· s2o ••ss1· 110 51243 eto 645&o 67o 1i'143 11ö eo1•1·

c.5.ti------4Jl.Q.4.1,.; __ - -- ''1-----····· -5'1:1.63-------- -~--6-ft-··------64-0'.3 -67t· _,,_,.&---·-···----·'1ti· 8Ö98t s2a Joott: ''2 57363 6tt· 65!17 672 '11'101 ''2 81111' 523 -5.0.-J,$.1: 1-73 17555 ·'.fl: .,,~,. 613 -7t9:J!J'-·······-- 7!3 81113 !24 '8lD1: 114· 57147 614· 65!17 674 73213: '124 8l433 525 ___ ;D...!U:__ _ ___ .J!?!L ~!11,1... --ifB-'------6.!!26.7- ''' - -.,.__.., - -----c~-1!1~ --- 8!-,oa: 526 5068J ~76 ,,,,, -~6 65199 ''' '1~511• 116 8.01'1'

.... 52.7 _ . __ fll.6Jl5~---- __ .J21 ___ _ --118'73-.... - 60--1----------~'5-lf-5---- '-'' ·1SUI-- ----- --,11 81tJ-'1-528 50733 178 SI077 &tl 15161 6,8 '1.111· '128 8t2o!· !529 'D IJU.. t!19 !.126.9 .... 6-62-0$ 6'19 '14tU 119- · ··- 31110' 535 J1219 teo 51365 '~o 86539 ••o 74269 1!6 81663 531· . '1435_:_ _ __ !J8.t.:._ _____ II11D1 6Si 6.6539--~------6.1.1:..--~ --·1ifó9 -731- -~--!H!Jt,·------532 ;1483: ,.2 588t3 632 6663! 682 146!!13 1:32 81f!1: _Sl~L 516!1. . .. ,_8J ______ ~-UII3~- . 6Î3: 6.6115-- -----~----,11------ ,._,,,, ,33 8!11.-t---534 flB&7 184 sto85 634 &7D43 614 148ts '34 8ssoo

..... 5..3.!t.... ,1861'. IJ! . 51313-~ . ,,, 67213 ,., ,,~,, . 13! 8310·· S36 52ó3S: ;86 50!89 &36 d7283 686 ''1!411· ,36 83647.

____ _!31 -- .52.1;19 J8'1 .JJ'7!'1- 6S1 674!1 687 ---1.j4JS,c 73; 8388'' 538 52299 188 59813 6ÎI 6769t 618 7i•4! 738 841!7' .93_9____ _,i.!t.5· . ,.. -64005----------- ,,.. .. ... t-76-9! .. . 619- -~'-J9fJ: 739 8424, 546 515!5' ••o 6Q245· &~ö 67~1' ••o t6tt7· 7•6 8443f 541 -- .. J2.6J!S. ,,1 6.02-45-. ·64t.· 680!It· 601 '16237 141 8411, 542 52923: 192 6Q269 6il· 88243 602 ,,4Ö! 742 8481~: 5_43" ____ , '!iJ_]~ ,,3. . .. 6Q!J.J7. ----643-:---- -6&1~5 613 '7Á60!-c------- -7•3- . 8481$: 544 529'1' .,. 60041 6i4 68501 604 '76813 144 8496,

. 545 ___ _2_13"55~--- . _J_i!!J· . 61031 -----US.._-·----·-6-81-09 - 6t5 7l813~---·-J•,_. - 8!1:89· 546 53643: 196 61205 ·~· 6880t 606 ,.,,, ,.6 884,1· 547 53.1U:~ - f97 61149 · 6~7 · -l-380t · 69'1 'l&-JJ. ·· .,_.,. 8!6-87 548 53811: 198 61445 641 68999 698 1'?!65 1 148 8,,,,

-·----- . -··--······--~ ---------------- !549 ... -''1-'1:.- ------Ü-'---~ -61445 .... ····-~- --60-1-39-------· -~·· - '7f6-tl _; '-4-9 -- --86itt'· 555 5t3S9 •~o 6t565 6•o 6'''' 16o ''6bl ,,5 e''o'

Page 46: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

4.3.~ . _____ f'i ..... A3. . ... N: A3 Jt. .. ~&3~- . N .3. ·

ae4o ,. __ JJU..:... ____ 9lO~.?- _____ .lf! _____ .tl~t~•- tä1 _____ itb6t _____ _9!1i:... __ fu••• !52 86-4óf;- ----- &02 91103 ·~~- tó4191 tö2 u.s,ót· 91~- 12306!1

_z,~ _ -~~-~~1 __________ .to:L ______ ,_,~ll ____ a!_3: __ .tö44tl_ oiu _ ... ü.i?os· __ o!:s... te3••.t· 7!54 86791: ao• ''''' e•• tö4t43 oft4 ttsebt 9!14 123&oo

_ !'! . 17Q?~· 105. tJ'169 8!11: 1Q.4~43 9Ö!I it40119 OS! t2300f'

_ _',?_!i :mt m .. ::m, ;r; . JUm !1!. l~. ~;;,;. --!;! -- ~!;;;,;_ :_ :;~o 976!J5:- - -1:08 - -----9-6t1? o8o •u7:~~~"'.r. ll'uo ;&..., .. v._ ll';llo •~•cc:o

·7!59 ~76~5 1 lol ____ f~l.1._ 119 _1ó,f.IJ ____ OQO -~J.f11t .9!19~ 12428t ;165 87?03: 11o 96o&o a-o tii!"'' oio t1,11~3: t6o 12•28t · -~:~ ::l:r -:tr- :~;=~ ::i: i~:~:~ :ii tti~i:: ::;- i::::t 763 8841U él3 973?1 1'3 _10~---~-::5:____ _ 91JL .. 11!10.1. ____________ 963 ...... t.2!03t __

-- 164 --i~h-jf:··-------.1·4-~----.,.,.i: .. ,.--- tó6239 914 11Î031 964 121183' --------------------- --~-'--- ____ _! 8 6 !!-. _________ ,a.'~' -----' 1!~1___________ ___.ii:._ __ J.i~--~~1. -- --------- -'~'- _______ 11~11 ,_ --- -------'-'.!'---------12.! '-11 ..... ·- --

-~------------ '1 66 889!1: 84.6 91869 866 1060!Jt 916 1tAI49 966 128999

--- l~l- ---:::;-t-- ---- --:1~ ---lià{~-- :tr ~l~il} :f: -tt:::: ::: · i::~::: '1 6 9 . .. _ 80 t99 I~ 9 ____ f_l.U_9 ______________ 8At ______ 1Ö.7Jil1~------ ______ 9.1'9 ftJ6.11 ... __ .9it9. -1 264tt ----------------- ----------------------------------------l,-o ----- ·-af,--e-3~--------.-;--o------ ••sas eto tà'1J8:s 92o t~••21 .,ti t26163 _l_rt. . --•~J~r. ___________ tt1.:. _______ ,tt.:'l4~~ _ . ..t,t:___ 1Q.72.e.J __ _ __ o.21 ..... 11~1•1 _,,t 1ett2s 112 s•''': 822 98985' e!a· 1g1oo:s 922 1t!3!7 972 t2'1ot1

.... l'!3 . 90087 113 9.1915 8?3 108S4i! 923- ~1,611. 9'13 11117Î 774 904tt7· 824 9taas e'• tóa:ss:s 924 1t16t7 974 ta'1&os

.' '~'-5 .. 9.~_0_4 ___ 4_1_' --- ----1.2~l _______________________ ,_,_, __ ,_1 ------------''-~--- ·- '"' --1-4"86 '-2.---- .9-1-5-~--- t.t .. ;_,_,, ---- _____ " _______ _ g_,_,_,____ !2-'-60~-----7~6 99687 126 90'105' •f• tij8?67 926 11.261 ,,, 12767fi '17.1. ..... .. i0 .. .!.1 ... ~: --------117 ....... 91873. . .... 8'17' .... 10!887-- 917 U,826S --- ~ 077 12,911' 11a 9tó47 aas 99873 ë;8 1ott27 o2s tt*e•• 9'18 t2A2os 71.9 i1!!7 .. QO !lOQt-37' tlo tQOt27 9-l~ tt .. 7tt 979 128301; 780, 9t383. 130 1003?'1 GfO 1Q0t27 930 l1~0Ö9 98Ö 121163 .

.. ..:.'1.8!.:. --- .. 916!3~ -111. . 10-0 377 ... -- - -fJ1 -4-Q-96.07---- -- OJ-1 -U.-0111-- -- · -----981 1'2192S 7Bm 91911: &32 100401 eet· t09,3t 932 11•!'1 012 t29Q43: 783. _ !19..1.1:... ________ _A_3.3.c. ....... 1110~1i. __________ U3"-----110.003"------·---.l33. 110.513'---- .. 0.13 ····---1'29.041-

-----~·-·-----~------ '184- 91965' 834 100919 .,4 11,0t05 0:14 t1,81! 084 1201SJ' .. ... ·-- .. . .. .:..'1..6..5.... .. .. .!lill~----··-···--.l-3! .............. -1.0.1-4~t3... - ...... -110-483. ' ' 9-.J-1 . .,.10111' •• ,. .,~,~'''

~86 92349' a36 tQ1313 e•• t~o69t t36 ~a§otts. 986 t3qoos· 7 8 1 ·_ _ --~ tJ_6_~--~·-----J_U ___ ..J.JU .. U.S.___ .---8U.:.-~-UD-6U----------~U-1----- -ttU-81---------- --U'1- ---iS0-1-0J.

--------·--·-----------~-----------------,88- - fase9· e:se ~()~673 éáe iS.o843 938 t2Ö4J3 988 t3dttf' -----·-----·--~la9 ... _ _ ___ lti.'lL __________ l3.2~. -----~-Gt.61_3c...... -·.!i.O.: __ . -1-t.U3.5c___ -·······- -91 9--···-llf6tl·--·· ______ cf.I!--~----.3HI''

------------------· 79Ö 93t6J; 140 101769 800 111123 9-tO 110713 990 13!.010 79! 93t.,: 141 1011.43 .. 401· . 1t1!J30 941 ttfi61 9ti- 13tO!J9· jj~ js~éj~ ••2 toaass 8~2 111539 942 i2t4ó9 ota tsit!JS· _7_93. P~3at: _ .1.43 1023.99 8-93 1~1875 -043. . ~IÏ40-t--- -- -993 13~29~ 194 i38SJ: 144 102•11 8~4 tt221t 944 11&411· 904 131401 79_5 _ ___ 9~.981: ·- l45... 102.1:11· 80! 112-211 94! -11(;861, - oos - t:U.61~ 196 9:1981: 146 101311 806 112289 946 t2'0., 996 t3t827 .'Z~'l. '4341: a4? to:s:u.t ' '' ts.24!1t ,., 1lf17?' oo1 t31tOf 1

198 94533: 848 103383 808- 11259! 948 121311' 998 13130, 799 .... ·~'tt~·--- !.49.___ . .1.03'119 ........... _ 809.. 112-R:tt o49 . - tti46!1 -9t9 13tso-J· ---- --- á1ro - -- 9 4 617 , 11 0 1 0 J 0 !J 9 9 Ö 0 1t3 a St. 9 !J 0 t 2 Ï 9 6 9 · 1 0 0 Ö 13 2 4 !J 1-

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··--·~-·-···- --·-·--· ······--· -·-··--·-- -----·-----·····---

···········-·· ······~··········-···

Page 48: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

- N A3 ... N.

··--- -·· --··· -·--- ····-- -------~--- -~-----------~------

Page 49: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

A3 .. il

·· ----- ----------i~~~------1-tló:~------llli- --~~tii' i:·~ i ~:~=~~- · · i::i -::t:i;.- i;ti· __ ---:::::r ·litf ---~::~t~.- il~~ ~:t:-~~ tt~r ~::-::} ·i:~~ - · ::tftr -t~~!: -·- :::;;i· i,6, 244593: 1f55 ~~1l3t I6ÖS 269489 16!5 21itl3 17Ö5 29'''~ isöi · ~••eet: 1t!6 25?295 ''~6· 269125 1616 a&jll3: 17ó6 195667' i5Ö7 2449'17' 1t!J7 257?99 .... .14ö1 209125. ..... 1617... llt!J4J· 11ó'1 .... 29!190'1:

···· i5öä 2i&t6; Il58 · i579-43~ i6ö e 210 at 1 1618 ae~o4o 1708 29!1959 i5Q9 . 24564?· .. .. 11!19 2,~,~~ .ilÖ.9~ . .21.0313 1619.. _ 11!2-41:. --11-09 .!964!!10' ··--·-·-· ----··-··-··-···15io fi!fe-41:-- -- ii6o 2stt31J !tic 21o1:~7 166a aasa•1 111o t9669t·

... 1'ii 2~5841· 1!f61 . 2fJ~19 j.6i1: 21.1D97 1661 18f8Î 7' 1 1tt· 206699 ~S12 246o33. 1f62 2!1815 ~6i2 211!4!1 1662 284011 1712 19677t·

--··-~····- --·--·····-------------.1'1_3 ____ l~-~-22J~------_19ft3~· ____ t5tJP9. .. _ .. ----1ID~ -·--2~164.9 .... ··-···--16.63... .a.aio.l7' .t.'i:s.: __ . ·2971Q.S: 15i4 ~46815' 1t64 298999 i614 211985 1664 ··~'*' 1714 297·4~:

-----~-·-~--·----------·--·-----------~----~~----_i_5.f.L ______ t47t1l; __ U6.! .. _______ .. JI9 .. Sl!5_ .. ______ f6[J_, _____ :.U!UlAJ.:.... _______ -1-U-1---184761'---·----i'1..J-----!9-'1111'·-1116 ~4?18§· tt66 asot11· iii• 212s1s 1666 aaio•t 1'116 t98ot•· l!i7 !4tJA!~ t!A.7 J!t.tu. . t't'1 .IU-4.89- !-647 t8i3fJI, 1'1:.,__ '-298ttt tsie 248Q4~ '''' t6Qót9· iti' 272633 1668 aei401 '''e 298753:

_____ ........... -- --~-------f5...1JL ....... .Jj!U0-~--~1f6J. . -- uo.sts~----- ----16!-.f.:.... ____ ..a.,.asaos------· 1-6~9-----····t&t&öt-- -- ·-· ~'1-9··- -a-t8,trs 1s2o t48ó40 tt1o 26óS95 !tio 27320t 16'o t86t41 1120 198to1· i52i . . .,24.8.43.1.--------1--f'!.--- .. 261-00J:_ ---- ---~tÎ!' -- ~t-73'-14 -- 1671 ••• , ... ,.. ·t7t1 1991!!t· t~22 24&679· tfra 26!1'' ~•wa: 173,as 1672 •••2•1· 1'22 t9t&1t 1ea:s:.. a•ae.41· tt'1s. 2,.,.,1, ,.,,.. ·273t8s· t&'s ta6•as· t1t3 ,,,,,,, 15~4 249087 ,,,4 262019 1614· ·273917 1674 11'120!' 1724 299939·

.152!!1 _ 249.41.. 151!!L. ~ .2620!9 tfÎI 1745-13 - - 161!!1- 28Î~6t ~'115- IÖÖ4!.9' !126 2JOQ~3: 1'76 262179 1&16 214793 1676 181585 1726 30070'' -~5i7 ---- .25.1 0!.3· ---i-tl7 - t6a5S.5· .. lft-7· --- --11-4U1- -1617 18+7'1 1111 ·- SÖ0701: 1!528 258119 1178 262707 1.,. 114t6t· 1618 18.011' 1118 3QÓ139

. .1529 2'87.~.1 1f~9 1.6.2.123. i6.1t 27!51.6!- 16'9 Ul~t' 1119 -3610!7' i535 251115! 1B80 263!!5 it.Ó 27,197 16f0 18.ti3 113Ö 3ij14S9· t53i 151.4~~~-- .111! 1631.1.15 1611.' .27.5.!17-~- . 1681 .... •••••• -·1-'-Ji .... --3ó-i6!J1' 1531 IJ143~ 1112 163787 i6~2 275t4! 1682 18.731 1732 3öi791'

. ---- ··-· -~~}~------l~i~~~·--------i!:{ -:!:i:~----·--- .. ·1::! ······--~~:::-~ - ----i:::- .. -- :::~-:i· . --t~·;:-~--:~::::·· ..... 1.53..5. . .. 2.l19J1. ............. 1f .. 85~ 264147 il.l!· 2'i1lt7 1615 . IIO,il !135 JÖI689•

1536 a't9tt· 1t86 164675 f6S6 a17as• 16&6 !95243. 1136 SÖ2t41 __ i5_3,_ ...... _2_f21.U; _______ -1JS.1~ ..... .2.6Ul.5 --- --·--Ï-t31:__ _____ J.'1Z.;4-!5---------- 4-68·1~ -··---~t-9 ~243~----· ·--t'13-1 ------ :J-031-8.,-·

tsJa aJ27tt: s.tas 1&4947 iele. 27Bt77 16&8 atö363 t138 sos''' --··--··------ __ f,_39_ .. ____ a.f .. 3tU·-------1l8.9 .... -t6!.5lt- ·····tf~-- -2-JStU---------····16&9........ t9i.,9t· 1139 -1Ö38St·

!545 as3zt5• t9to 165955 i•~o 1?8369 160o tti1o7 t746 36395' i54i 2,:5 .. 6.4.1 1191 26!9!5 !641 278101 1601 lt{JIO' t 14i 304!6' i542 t53BS7 t~92 266105 ~6~2 at9a6S t6oe 2'1'~' 11•e 364795 1

1543 ... 2538~1... 11'3 1661!1.7 -1643: . 1193QB-- - -1603 lttiO! -!14:5 3047t!J' i544 254t27. !'94 166,95 i6i4- 2?9449 1604 ~tj41? !~44 3Ö416t' 154!'J 2J44!..5 1 t195 267l79 t645 17-9641 1601 29t4!11 1145 301341' 1146 2J4823· 1-96 267179 16At 210t69 160. 29t4t9 1746 Sö6o~? Ï!41 . !S~111:. 119'1 26'134? i6~7· 210!:69 · !601 !f.l!-f' 1141 :JQ6!A1 1548 2552~1 tt98 161731 !648 210!69 1608 10301, 1148 3ö642, ..

.. .. 1549 25!'4.17: -1199. . 267?31- t640 28.0'4' 160.9- ·····~;' 1149 3Ö6-9tl' i555 2JStt9 1&00 261761 {6jÖ 281225 1150 29S~I1 11e6 3Ö719S

... ·--·-·· ··-· ···-··-··--------·---- .....

Page 50: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

.. A3:. ... !3 .. N

-~--~.~-··-~·-·--·-··--·-····· ..................... J.!St ......... ~.illll.l:~ _____ 11.01 .. 32.002.7 ... ____ . .i.lfl.:---·---333Jl5 1901- !t.tUL ... ~---t!lll-·· 36ÎI69' -1752 3D729t· 1102 3204.6t· 1112 333115 1902 34èOI7 1952 36~389·

-~-----~----- __________ ll~;:t ___ ---~O_{~tU.~-----·J.l.U. 3.iU6 .. fJ3 .... ill.!. _______ 3~~4!.~7 1903 . _. 34JO-If _______ UJI3----- !6t86t i754 307981' 1104 320797 t8i4· 334867 1904 34f017 1054 361101' .11'5 30837.1.:. ..Ut0.5 32iSD1. lil!' 3!J48.6i ttê! 34J:U.s 1-01.5' S6f493; 1756 308311: 1106 321,81 i8;6 334939 19Ö6 S4.all!· 1916 3627S3

------·-·-----·-----~-----------·--·-·--·---------~ 7 __ ,_7 ___ ~D~Ill _______ 11D.1 .... ___ :121~81___ ___ .. iéif~--- ~3J,.t!.9 ______ .. 11Öl' _ .. 3.i86-' --·- __ ... --tt!Ji------Uitt-8'---758 308947' 1toe 321877 iai8 3~5419 195& s•~o83 1t1e 3633ot

·--·------------·------· _____ ------------------------------.17.59 _ :"ol9-t_l· ___ .. _ _1.1Jl.i_ ____ :.\22~.t3 ---iáÎt: ______ u!'l!t .. _____ t_oêi.----34Ö4-tt --- -- -tt19----------S63Uo~-t76ö 30tótt. 1110 321&93 !l6ê :ssS97t 19~o :ssgot1 196ij 3634'!'

. 1.!6.1, 3093.31.:. 111--1------.J2324!L t-1-6&: 316-42-' t•t1 ~ffit•l ~!,--96t· 36316!..-!762 3096t9· 1812 3233&9 i•62 336t:Js t9i2 3Sot81 1962 3&4mtt

. ~- ~ . ~- ----------i76:L . ~OJI~I)J' --J._ill ___ .H.l&Jt.5~------ !!63 . 336.1!1~-----~--~U-Î-3--- --JfÖ-'i' -10.3 S64J41L~~------ --i764 3101,7' 1114 324!1? !864- 337017 19{4 SJittS 1964 364581·

~~ ~~--------------~--i7..6!5_ ~--~31f.t411_ 1&J.5 ______ _3J_4t-J.f____ ---~!•--'•· 31"16 0.3 . -1-9~1-S.------SJi-3-S9'. ~196J -- -- 36,0-6-t: i766 311dt7 1116 324325' !166 337,81 19i6 3St:S3t· 1966 361493:

... i'i&1 . 311nt7 . 1111 324.803 i •• ,. 33Bto7 ttt'l 3Ji713- 1961 36t4•3-i76S 31111!: 1118 321!65 it,a 3382'5 19{8 SJt91S 1968 36SI4t·

· ··········------~-~tjn----~---Jtli:t---~·---···-i{i~ ·--··········::::~i~----------ttt:----- -:;~::;·· i:l: ::t:~t-~ · t:;s ~::::;: .. i77i -~1~~~1: ______ .1111 ...... 3218.31~~ ___ .ill1-. ..... l38147 .19tt· .31t4~U~~-- ~- .--~ 1911· .366'180·

'"f72 312.,3 1822 326029 i•?2 339167 1912 31~839· 1912 36693~. 1113 :u~Soc,~: 112.3. 326029 111.:s. 33t2tt ttt:s 35S079 1973 3614:U' t774 313a,3. tt&4 3•6125 ief• 339183 t914 3S~27t 1974 367811

. . 1115 .. ___ _31~.22.3,_ _______ .ti2.!L 326.4.61_ .... ····~-----ili-'-'· .. 34-0t.7.t ... ~-···· 1915 . 3tS943~--- -~---19--!S-' . 3-61111· 1776 3132'3· 1ÏI6 327113 i&16 340371 1916 35.303 1976 361~1'

... 3.1'-'- ---~-·~ .:S13.!6t: ________ tJa7 _ --32-'-'-21- ~· ... iaJ'-··· --3-4-0-''-'--- 1o11~ ... ·3si:sos: -~-lt-'11~----· ---3·61J8o·--~ f778 31-d41: 1118 327!17 i•t• 341j67 1918 3jil43: 19,8 368~33: 1779 !1~281. -Uat.. 327997 it7~ 3•U,06~ 1919 3lfi71J 1919 :J6J28!J' 178Ö 314177 1130 328285 1è80 341307 19!0 3191Ï'' 1980 369181

-·-··· __ ··--~-~------- .. ~.1181 . 31,193; .. 18.31-: .... 32!285• i.àt. 342t2'1~---·-------t--93-l- -SIÎ6t:S• ·191i ·· 3601~1·-·· 1782 311601 1é:s2 32&50? ie~t: 3422~~ 10s2 3ti7to 1t12 3608ts

-----~-~----i7a.:s .... :stl_6o.t: taJ:s.:____ . .3.tfte.as_ taa3· . :S42J'1t ... -~ ---tts:s.-- ssio:as· t91:s ·360&t--s: 1784 315985• 1&34 329311 iäl• 342179 1934 386119 1014 369813•

-~ .. _t'l8! 31616; 113! 3215!7 !811 3421'1t 10:J!!IJ ,,,,,,, 1985 37Ö293 1786 31&609 1136 , •• ,.9 ~él6 343539 10:16 ,,,,,, 1916 370860

.. _________ ._iie.,_ . 31&77.7 1837 33Q031 ---i-187 . s4:ss:so ---·1--93'1~----- :st;ais• ·1911 . 3'1103'1 i788 31&717 1838 330373 ii••· 3436tt 1938 3lf!J07 1988 31t311

. j~7.at__ !1.1óJt9.:_ .t.a.so s:n~:s7~- ·-- i-aio·-- 344471 t9:J9- -:ssj-?At--~-~~ 1989 :ntoos.· i795 317711: 1140 330421 iltÓ 344859 1940 !f~OI9 1995 3'1!8t· 1'191 3177!.3 1141 ~1.30001 ,.... 34!5899 .. 1941 3Jt3l'1 190f 371189· 1?'92 31771!· 1842 331093 1802" 341243 1942 3JÀ58!• 1992 311!8!1· t793. .:JUh.,'--: .. 1~43 33!23'1 ia0-3~- 345483 1943 3tll8t·- -1993 372173 1794 31SSI9· 1144 331597 Ï804 345843 1044 3JÁ803 1904 37322! 1 79!) ___ --~-~~~lll: ________ ~J.a.~_,c .. 33;0?'7 · .1805;__ _ 3f, __ 84.3 1945 3J0015-' - -----19.95: ''3413 {j~i- 318961· tè46 331~13 tao6 346i&3 1946 31~9öt 1906 3734aj i1_,.,___ 3l~at4J ~~ .1~~~' 3314.13 !ao?' . 34627!. 1o.t1 3&äot:s. s.9t1 . 3'3t8t 11oa 319489 1148 :t3aso9 ~e•• ''''11· t948 :t6go9a: t9oe ,,,,,,.

_________ ~~-------1~"'~'-- .. :u.!~-~-'-~~- ·----~..1.L4.'-~·----Jaa.779 .tst.•~ .... -----34.&.t7t.: ........ --~--tt4G ..... :s1o • 3-3:__ --1'''~--- .... ,,_.,,, ... . iaöö 319741: teso 3333'19 ioöo 34709t t91o 36~269· 2öóö ''41bo

.... --- - ------ --·-- ... ~---------~·-···--··- --~---~-·--- ·------- --~-----·---··- . ~- . ··------ .. --- . ---·-··--·--------~----~-

Page 51: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

Al N... ~~L _ N A~ N .......... A3 N . . AS:

~0 01 3 1S28 s~ . _ -~~~l ______ ~t9..1-'9. !lÖ1 4Ó~!UJ.. 2.1J.1: 41Î2tt 2201 . 432913.- - .... ................. ....................... _______ .. ____ ~-------2oo2 .. · · 3784'' 21s2 :set3!J1 !i§•· 4§Ja'' 21;2 •s.•~·~- 2202 .tJ:n.!l~

~Ö 03 ~l5_693: aJil· .......... JlU!1~7.. . .. I" ~'~3: . 4~'~3.19JL .......... .21..!3-._. 4&. ,H_ ~ ·--22~3: ....... AUt't:.. · ··--·-··-··-·· ................ ___________________ .. _ ·- ........ 2oo·4 3 ''as'. a is 4 3' o 4 :s1 a i g • 4 9 4 o ~ 9 aH• 4 41 ~a, s 12 ó 4 , :s 3 2, :~

~gg!!J 3~:~~9 '~,,, ~:~::~ . 'iO.' ·~4.61!... 111!5 41~501 22Q5'. . 43:17'1~ -----------------.. -------~-------· :~ ~;. __ ~I,:~_r ______ .. ~j;J__ .. !f9t1J'1 ... :til· .. ----~~:~: ______ :i:; :ii~l: .. ____ -~=tt ______ :::i:t-.... 2ooe·· · -:J7'1ots: tri se 39153!5 tu1a •o'4'9 21!18 •eoa•s· 22oe 4342!1'1'

1~~~- =;~ti-~L-------~l:~--- --~:!:~r----- ---:ti:-·--"- :l~-:::--····· :i:: -- ·::t::t------:::-ij~----!::::l' 2-Q1 •. 3782117' 2,.61 ~~~~'1 2J. 11: 4Ö~t9!, 2161 ••. 567'1 121.1.' 43560'' 2oii 37823? !i62 39161!5 tlta 41)6343 2162 •ai349 2212 43!'7os·

........ ······--·-- .. 2.it1_~ ___ 1I!.4.1.!' _____________ at•.3.. . ..... 392.t1,' .......... -t!ia ............. 4~6.J.!s.. 2163 •ti:!IA.s.:. .... ______ c.;a.t:s... 4:s•ao'· 2Ó14 378861· 2864 30J,J9 til• 4Ö71Q7 2164 4ti''1 22&4 436783:

··-····--··2.Öi-'._· .... !~.!~-~1~ .. --- -~j._f·_ ······ 3ll04.7 .............. tt.1!.-. ...... 4..07.441 .21-61 41!-.1.1.1 ...... _ ... 2215-.. ..436,13 .. aói6 :s79óo;: 2866 393719 211•· 4ó7I07 ~166 4ti5ö1· 2216 437041 20i1 ~79..1~~ 2161 !9!.11.:1... 2ti7 4fi8tl9 . 2!;67 4ttlii 2211 43,383 !ÖiS 379483· tf68 394015 ~~~~ 418269 2168 4tiP!I' 2218 437i91

.... ___ .......2.äu .... - 3.7..98.6.9.:._ --21-"-----.3t4Si-t-:---·-···-ftl~ 4§8169- ··· · ··· 2169-- ...... ·~~S09•······ · 221! 4312't' !Ö25 379965! 2f7o 391231 21'ó 4~8601 211D 41,'101 221g 438361" 20 21 :ne7.81: .. a.,l1------.:St!t.:S1--· ~~tl 4QOoas. -----21"'1··-- 4tfOI,········ 2211 43818' 2022 38tti69 2&72 395.2:5 2t22' 409301 2172 1$!4031' 2222 430201 2Q23 3lió69 . 2l73 305163 tiil· 4Q963'1 2173 41481'1' 2!13' 43920'' 2o24 :s&t4óS' a,,. 39!to3: 't'4 4p9113 2114 •eso41· aaa• 4391'' 202!5. 3ll9.1,.. .lt~~-----39.640 7:..... . -IS 91: ..... 4~-0.l-4*"~----21,5:.... ··41f04Jt ·221'- -· -:41J9,9t•·-··

~~~; ~j;_::J: __ , .. "_, :t~t ....... ::::&~ '.Jlt;:_ .. ,!t~::-... m:i;; '::s:::. •••• .:i:l.u• .::g::;; 2ó2s 312~91: aj7a 397079 ~~·8 •to•a• ~1'18 42j960· 22ae 44jl30 2Ö29 :sl~o~.9· aa7t '''b 79 at•• •u .. u.1 21 '' •ttt:J'1 . . ·221·9 ••1•1'' 203Ö 383§79· ateo 397127 tîso· 411965. t18o 42t!lt aasö ~4!65' ag3t· --~Jll1~---------2tlt. ...... 3978.47 .. ----fll-l'---- 4-t-2J.7.7. 2111 . •-t•••:~~----·22-31-'----••~•s~· 2p32 383,79 lf82 398t35 èl-2 412411 2~82 ~1!211 2232 441943

.... 2Q33 ______ ~!.~i.U~. . 2113 ... 311!0.3' .. ----...!~3-- ---4!2997 tt13~ 41?2~..,: .......... --22-S-3 - 441131' 2034 384]95: 2884 301687 ~134 413111 2184 4271ól' 2234 442543

__ .. 2Ö.3.'-. . :,~~n1· .. aje!J . .lti.al:t i~"-._!1 •1:saa1 21as •a?,•s- 22s5 A4t&3t· 2Q36 384947 2886 399!19 ~ ~6 413621 2t86 41,2!1 2236 li428:S~-

.. ~g3J . ~85215~ 2j87. __ .39JIJJ.fi __ ~-----' -~!~ 413813 218.2.. __ ~ 41,.61!. 2237' 443t0t· - ... . 2Q38 381451· 2Q88 39~14,9 ,~~8 . 4!4117 2188 41~6~·- 2238 4438~.,.

2030 385~_!11: . 2JI9~ ..... 4.U.0.1l'l..... 2l!U 414509 .. 2119 ... -.-•11409 1239 44351' -·-·--· ·2o4o · - 388643: aito 4ooea\! ~i-b 4i4!1Jo9 21oo •U!t6•7 2a•ö 443'''

aö.4j.: . \!J~ dar 2i.9.1, ..... '~.o.a11. •i•t· •1544S 2111 ,,.,., 2241 444!,1' 2542 3866!7' 2i92 4ti0991 1!~2' 4(!5915 2102 .,.,.~, 2242 444343: 2043 _3,69~7. -.2j..9.3~-~--4.4-lt.!1. --ti-43. . .. 4t!ifl2!t .. -------2113--- Alfi~&s,: · 2243- ···· -44,1-ol·-2Ö44 316987 2j94 401807 li~4 4i6ó93 2104 C35468• 2244 444898• aq4s .3115'6.3~- --tits:. __ --4ol807 ~~~~· 4'·'•'-~-------ttn ~at·~· -· --aa•,···- ... .,,..,.,, __ 2046 381941 2196 401927 1146 4t68!3 2106 431201 221&6 44621''

:~:~ ~:~;~:-: ·· ~t=~ :~:::; · :ï::- :i;i;;· :i:: :n:;r ~~:: · ::::i:: ····-22-49 ......... 3J.83J1~ ........... 2J9 .. 9' 4g3a,OJ- ··· •1~,'- --4i?-t.-2fc.... 21!9 4IÏ''" ..... ··-·22*!~... 44.'191•

205Ö 388727 IJOO CQS343' 2180 418229 2200 43tt•!S' 2210 44'1111'

Page 52: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

A3: .. N .. ~.3.:. ~ .. N A3 .. N .A3 N. Al:

441611 9' 2101 462!5!59 . t~f.1' 411 -~~- ______ .2.411.: ____ --~-ti5.61 2411-~ __ !ïL8S_63.L .... -447aos~ ·2Toa··----.&i7J_i____ t382· 471ott t452 ••~o•' t482· sfie683: 448311 2103 462,,1 .. t:f_@3 -~!111.3. ______ .. _2_4_53 ....... 4!~2,9· .. 24!13.~ .... 10..03.0!.·,_ •f48759 ---2 .. so.-----.-6t71i-- · · ······· · 2314 418'8' t•ö• .,,.,, 2414 !llöo6os: 448719 .... 2SftS 4.63069 2Îfl 4Îla.7i 24Ö!S 49~919 24115 1Öt691' 4(8jjf 2!06 46!837 2386 479265 2466 4946'9 2416 11ÖOOO·

~::-~:~~--------ll~l-------i::::-~ -:ti-r--- -~~-~-ii·:---- :-:~i·- · -- :::::;~·- ·------~:::- -----:t~::::· 44998!: . 2~09. 464917' .2~j, ____ 4l0t!9.. 2.409 49Î447.: .L4.1.t ... -l1111Î 1 -·-

4föi'!f_____ iito- ·· · 46st!o1 '*•o · 410369 241o .,,,,, t465 SttU''' ~!Q.~'!l: . 2.!11· .. 46130.1 2.3 .• 1: .. 41.010! 2411. 41.1tt. . ..... 246!.. S!.(58'' 411279 2!12 465505 ~~-~ 481113 2412 4963(1 2462 ''110'1

____ 4,_1.2.?1 ________ 2_!13 ....... 466.0 . .It ________ itÁ!.;__ j_l1.1!3 .. 14{3 .... -'''''-~----·--· ~·~~3. 11{9'1i .. 451639' 2114 466417 ti~4 411353 14i4 499463: 2464 8!~06''

-----··-------- __ __ _ _________________ .2tl.5 ____ 4_,_tQ!l~--------2l1.!.-.. ___ ,_U.6.14.2 ____________ tpl.' ...... 48.17.3'1 ---· 2415----· -40;-4-U---- --- · 24'4- --·ltt4J-t· 2266 4fi4S5 2116 461b41 ~j~6 412361 . 14i6 4ff463 2466 !lt311i·

.22-67 . 4527!9 2117.. 4AlU..9 ··- 11•1 412361 14{1 4tij801· 2461 !1t3!3-0 2268 :~:~~:- _ :.ut ____ ~:~i~t __________ JJ:t !:~::: .......... -J:tt ______ ~:u;: ... _ ::::. :i:::~--·-- . .. ---···-------------·-----:!-~~ --4li4i9!-- 2sao 461&!17 ~~to 413137 a•ao •••••'~ t475 !114401·

~;~~ :-:::i~> --u~~-----:~!i~ ·- -llti- · --::::;-; -~!t -- -:::;~; ::;i :i:::i;-- 2.2?! . ~f4ó!!9• 2123 4691!53 . 13!3: 414421.. 2413 4t;1S.9 I•U:S 5!481-J'

2274 4f-7!1: 2324 460489 ~~··· 414619 2424 49.863 2474 8!18ti: ---------------------------------- ______ 2275 4.'~.1-~.1: _________ .2J2§ _____ 4..6_91J!S._______ 2311· ........ 48461.1. ---------241!'L !10lS.t9 2.4'1-S ·-----B-l6-t-1l·---

2276 435335' 2126 470089 2~!6 485173 2416 10§841' 24?t !l162ti:

22 ~: ·----:,-cf.-~~------ :~:i·------:~~~~~·---- ---:1!:-~----:~=~-:------- -:::: 1~tt:;L --···-···- --~:;:~----··-:i1~=r• 2272 ... -~~-~l63; .. .tt2P 4'10665 23?!~ .4t.636t 24a,. 'Ol81!5' 247!. !l!U,Si! 22sa 456as5; 2ilo 471315 1310 486461 2430 !Ot479· 241o ss.1221

....... .22EIL ..... -~1~_21-''~_ ............ 2.131... ..471.865 ~~·1--~4.861tt. 2431 IO!J7-9· ... -----..2.4.81- ss. 'lees.· 2a82 456775' 2132 411165 '''a •a7aJ7 2432 so,sss.: 24a~ ss.etlt

_ 22el.. ___ J!~-~1.9..:.~----- 2;:s;s __ .. 41.12'13. _____ ... .2$83~- __ Ae7.a!l . ., .2423-- ... so.a'1-'l-'---~- .. 2413· 11.&6to. 2284 457o39· 2834 472101 ~~~4· 481421 t4S4 so•'''' 2414 ss.oóo~

.. 22as.. 4J.Z7!U. 2l:S!L . .,aso1 tîis 4a7oos t4J!l tos7ös: t4as 11oaot· 2286 4'&335· 2~36 472903 a386 4182&o 2436 so~eos 24&6 '1'''3

----------·-~ ____________ 22.6.7 _____ 4,~3.:55'. as:u __ 41..t3'-l----------2ttl·--- -.41817'1 -- t4:U-----c.Jo•t--i-t."···- ... ,,._ • .,~. Js,ot&l· 2288 4f8335 2S38 4?,665 t$.8 41871t 2438 jof44,. 2418 !12Q10;

____ • ______ 22.89 .. 4_fSl.l9... .28.3.9. ___ 47.4121- a:la9~ 419tto -2-439---- --104447-' 24&9 12063f' aa•o 431959 214o •?44ó9 ts•o · 4a9,os· 2440 •oi5!s.: •••a 'atót' 2221 419391: 2841 47.4.?69. IJ_,1. 419,0!9 2441 · I-OfS03: 2491 !!12130'1 2202 •'•sa!. 2142 4712'13 t~~~ ••••at 2442 10,817 2492 12tso1·

- 2293 4f~l51: .2!1!.3 .... ___ ,,._,,13 . 2303 40012-1-- ----2«3---· -!10·0'1 2493: 92~661' 2294 461663. 2S44 47,~8o •~~· 49124! 1444 •o-2,1 2494 se,oot· 229!1 4~JHt~3, __ . _____ uu ___ .. 416.!1;61. .. 23,!' . 411-4:.13"- 2445- !lCit6.1! 240!5 ... -82--.t.tJ? -2296 46t81J! 2846 476449 1396 491433 2446 !O,Itt 2496 !littO'' .2227 4.61.3.3.5: ... " 2!47 476169 1391 491117 1447 '''''S. 2497 922243' 229& 46&119 2••• '7'''' ~~~~ ••eost 2448 •ottJI a••a 'e212s. 2222 _____ .!!_l98_1-________ 2.t4.i . ___ 47?3.8!5 -~-·2-'H--------U-t.O!I-1-- - 2440 . 10?6!1' . ----24.0..!-- !123-111,. 2300 46!983. 1850 477165 14ÖÏ 492177 2410 !IOA371 2500 ,23301'

Page 53: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

---~-------~J _ ._.J\1 ......... A!-.-·--·-· __ . __ N. i3 tt - - 43~ -- N -AS-·

_2_5i1 ______ ~_2:Sfi._1.l_· _______ 21!5.1 ...... !540.113... 2.&ii1: '''''1--- 2AI!Jt .. l!irus.... --27 . .d1------!J811U:-- .. -------------· 25Q2 IJ214481: 2.!2 !I4Q!!I3 26Ö2 !1!15103 26!12 !11i36S 2702 !181730

··-······.2.5 .. 0.:! ... __ I!JJ~411. - 2fJ3 !14.06.4~- ·-· ·-- 2,43 !156213 26.!3---- ''~-'!ll,. -21Ó-3- ••• ,,. 2504 52.913: 2t54 !141097 26•4 I!J!1647S 2614 !171217 2104 IJ818t7'

. 25.Ö! s~53?3: tf5.5 !1411!.15. . tAKs '''''7 26!3 ''~al'. 11o" eao6es.· 2506 !125771' 2t!l6 !141185 1'§6 !1!17387 2616 571347 27Ö6 59Ó06!1 2!5Öl '~6U.E 21!1_7 !J4160.1. .2lO' _ _ !1!17317 26J7 -17'S,:U,: -I,Ó7 - J00233-

------ mi- m~:i :m i~m u u ---- =~m:- --m: ~nMt !Hi mm .t,~I 5271~~ .. 2..861 "43ljt: tAl! !!9J3t 2641 . ''ts.•• t7u. sos.s•t 2!112 527241· 2J62· 54!~49 16~2 !159499 2662 5'f48! 1712 !191601

~--------- 2'i_:,_ ~l~~~1L_~ ____ ..2_f.t_~c__ _____ ,4_1_1~l- . __ __ 2 At-3 _ __ _ !1_!9'1.8 '-- _ __ __ 26-63 _ __ '-''":s" .. ··---2l-~J:_ _____ !Jo.ttt-o.· -····------------------- .. 2!Ji4 52Sttl ,, 2J64 !14!929 16{4 560:539 26'4 ,,, •• ,. 2714 !192840·

---------····· ..... ~·-·---------~------~~---~-~-----·i~-~---·----;:: i:~ ' . ·-----.:!!~~-----l:-~·:i·i--·--------·:·:t:--------::~!j!---------!:::-··· ··-·--··:; ;::~~-------:-;t:l·----·:::::t:--- .. 2-!it !U.S.i6i . ~IJ.'l J.44J.t.J- __ -- at{,. - J610I9 · ··-· - 1667 ----f,ÎiJ-&· 2,1'1· !-9Sttt~~ ~518 529319' 2168 !J4f!05 1618 !161130 2668 5,1331· 2118 !103851'

_ --------2!iQ ....... .!tt.JaJ ___ ~-------U.4.9~-~--.!145•&o.. -- ---. -i&I-t--------S-6--t-*s:s--------- -- 2-669----···· -··· ' '•e••· --- --- --t-719- -- ''J&!J' -·- · 1121 sa'''~ a•?o !1460!' •itö !161023 t6'o sjt'''' t1t6 lt4ooi·

... a.!J21 5299-!S-------21-71--- ---54-61-!l.J- .... --t6t-1-"-·----J6266'1'-· ··· t-611 ,,,,,,_ ····· -· ·····2flt· - 194621 2122 53o3es~ 2~12 146415 ••ia !162915 a6t2 ''''~' 1112 1•••••, 2523 !!130633- 2173 !146953 · ltf! 162055 1613 S1,!bS t't3 lt!lt!IS: 2524 530633 2174 547&'1'3 2614 !163851 2614 51.!87 2724 IO!J30S: as as sJs4'3; - aj?s .. - !47&1:~. 26-ts· 1-63611 -16'' f&ott•t· ,,,,. ''''~'-2526 5321~3: 2,,, 141865 ~-·· !164819 26'6 58§235 2126 596!169

.. a121 531193. a.J7'1 -J4110" ~,,,_ -- 864147 - t67'1 s&o~~t••· '''' !196J6t· 2128 53228~ 2178 !141Qt? 2it8 !164517 2618 586979 2,28 1067!3: 2129 !5!.30fJ9 ,,,, !14-1101 tiit· . 5641·1!- '""' -ttft•'1-•· .,,,,. ,.,,,, 2535 533297 aseo 141ft3 26:to 56559' 26&o e&t1t3 2136 !1910!11 213i .. !J337d!~ 2181- ---!I4Q485 - !6~!.- !16!!559§ -2681 !lli98-1' .. . 2131··. !J98:Jt1· ~932 133945' 2~82 !190025 2.3.2 !165691 2612 58t34? 2732 59144t· 2533 ............ .!31.~7~- ......... .2.tl3.--. ---!li--0-0-2-5--.··-----~--~3·· 566-Q-'14· 2613 ..... ,., ···21-13-· . ··!1988-!J'. -·· 2534 !1348!17 1184 !150217 1634 !166107 2614 58t467' 2734 '''1•t· .til.! .... ~. -~J~4_e~~-l_: _______ a.ils_ .ss.:o&t!... ~-~-•--· '''''' ,,., ,,,,_,,! 2735 e9t!;,,., 2536 53~025: 2S86 !15tS69 2636 !1618!19 2616 583139 2736 199281· 2.531 . ~lr& . .l4.f.' . ..21.8l ..... !l51!1'-~······ ...... --àll1: ......... !1677-Ó-7- ·········-----2.617-- ... --··~t-10•------ -21~'Î-·---- -590473;-!!138 !136t29 2'88 ''1161 2631 561947 2618 !18S111 2'38 !19994t 2_,3,____ '--'~3-~ __ s_: _.2tt9 _____ s.!tt4s _ .t.&!t __ !,,,_._, -2••• .!las•''' '''' ... •oo:ta•· 2145 5J6&33 2J9o 551233 ~••o S68o43 ~69o ,a•t•si a1•o •go•••· 2'~1 '-~~1~1: aj9.1 ssaas.3.. t64s, 56828f 2611 ••''''- . a7•t· •ott61· 2542 5373!3: 2192 !52341 26·2· 568907 ~692 !18.9i!J 2142 6Öt149 254:.. !37.3!U: __ .. 2193 s'a'2' -•••~ s••t47 2603 '''•t•·- 1,,3 •ot••• 2544 537353 2194 553~!7 t644 !169363 2604 186tS9 2'44 6ötf33: 2545 537641: 2J95 553.45 26.5- 569915 2605 !186tJf: 2745 60I!J09· 2546 538409 2~96 553685 26.6 170587 2606 58641~' 2146 6öt96!1 a!1u7 5!!7.69 21•1 554333- t647· ''o'•' 1607 lt67tt· 2747 6Á33t~ a548 s3s~s7· 2tt8 sst621 e6~a ''osst t6oe 'etoö:s- . 2'•e 6~3397

_ ... 254.9 ... !U977.7 .2899 !54611 26f9 !71319-- 26!9 ~1!363'- · 214! · 6QS901' 2555 540ttl e•oo 554911 2610 571139 21oo ,.,,., 111o 604711·

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... N ..... .tL N . AS

27!51_ 6047&5; 2•01. 621013· 28Î1 ·--~~151~1 ......... ...2.9~UC .. ~---''i9!J . 2911. .... 6111.4~: ______ _ ·-------- · ·· · · ··~ -----~-~--------ri75_2___ · & o 41 s 9 ·. · · ···· · ·21-o2-:-~~-~iif!fói-- ·· - -a 8 i 2~ ··- 6 3 811' 2 oH 2 'si 1• 3 2 o" 2, ' ' t" s o,

i-~~!--·· ~-~=i~}------==-~~~---- ::t:·};--~ -~:i:c ··---1:-~:~~~------~--~B:'----:JI~#~- ······ :::r- ··mt~;::-;:--2l_,, 6o ~1~3:. a.tJt5, 622501 2a;s .639tl'i toa! 6!1jttts· atss~ 6JJtt5' 2156 6066t3: 2806 622933 2886 639418 2056 616487 2916 673245

--·--~--··· -------------~----~-~ _____ f'Z-'.'1-. . ... ~O~_li4.1~----------2.80.1 ____ .. 622.133 __ -· ... tJ;i ...... clU.t1S ....... - !.951. - . .AI-11!k - ...... ~29fc7 ....... 61Jl93 -2758 6014!9' 2108 623l2!5 2888 640!7t 20Ö8 616771' 2918 6?42??'

.2'Z5tt IIJJ.l.!21 __________ .!Jo.o ....... .623•.4.~t .. ~ ----~U!J ____ .6U6.Q3,.. 29ät· 61;65.5' ------291-0- ----6,4Q,7-- -- . !?65 607621' 2110 623023 té&o 64o&o9 2o1o tt5lotl' t96Ö ,,,.,, ~'Z61 ~~!.9'-7 211.1~ 624307 t.t61-- 641.''-1 21!;1 &Jiots. 2-'ltti~ 6'1S3:t:s: 2162 60826~ 2112 624307 28~2 642D67 29(2 65f2:JO· 2062 611821·

....... 2'1~.3 ----~oe~~p~---·-····~ .2.813 .. _ ... 62,027. .. _____ .-!J6_S _____ 6_4..2lA7_ 2913.. .. . ... 6!11.!l!.:.-~-~-.!.i63 .......... 6'J!l8S1' 2?6• 6087~9 2114 61f!07 ti~4 642t17 29i4 ,, •• ,3. 1964 616121

.............. ~ . . .. _______ .2 1 6_5__ ___ ~J-!.4J!.1.: _______ ... 1.11.5 ___ ..... 6.2IIO.l __ ~---UA! ...... ~--'-'-21'11 .. 29i!L ........... Alt.4.!l~--------1t.6J -· ... ''63651

2166 6097!l7 2116 621!31 2è~6 643D!l1 29{6 6!1~7!l'J' 2966 67,39j .. 2!6'1 611'''' 2J1.7. .6260.11. 2$i7' 643339 29{7 66~011: 2967 -67730'' 2768 609925 2118 626211 2á68 643JSt 2918 660419 2968 677401: 2769 610405' -~·2fll._ ____ i2.6.l!l~.- 2.8.69. 6.4811. Hf9 ...... 66Ö4JI 2960· 6'1111,

----------~-2-fir· &ioi:fr 2120 627~43 ~•?~ 644683 a9tó ,,~,,:s 201o ,,.,os: ·---·--···--····--··· . 2J7i . 61$4~1:. _____ _11.2.1_· --~62'1~U'l !1?1:. _ 644.613 __ UJi . A6~34t· 2971 618'141'-

2772 611749' 2122 628003 21?2 644827 2922 66t8~9 2912 67.1'16~' . 21?.3. 611.941:... 21.13 621003 . 2$73 .6.454!lt .2913 6610!.3 2113 6'101lU.'· 2774 612613: 2J24 628291 ,.,. 64!J03t 2914 66é30t 29?4 619611

_ ~ t? !!t ~ . 61~t.11. _______ u_a_,_____ _,a_e 96:s .. a a ï' '. -·· ____ ,_t6.tt.9______ __at.2 '- . .. . . '6:ts.•t: · 2 ' 's. .. . -'' t6 ,., · _ 2776 612733 2826 620809 2816 646219 2926 66S61t: 2976 6?t82t

.................................. -.~··--···--···---·--·----···---····--·--- -····~·-·····-- .2.lf.1 __ ___ .1 L'.U.l ________ .tt27 ....... ..6~.Q001. . . - - . 2$l7. ......... 6f.6..699 __ ~ . -~-291..7..... . .. 6.6l611' .... .29.7.7 .. : ...... 68.ÖS.~·· a11a 61364' 1 ata8 630235 aefe 64689t 2928 66s6a1· 2918 ••beat· 2779 . 6i~_i81:.. .. .1129 6!06.19 aa;o .. . 64Aa9.t . 2929 66.1ó1· 20.70 681189 278Ö 613981: 2830 630819 ,,.0 646163 2930 66.431 2986 681381·

......... 2.!~.1 ..... --~1.~.l~_l_' ______ .. 2831 6.308'9 - ---.2·~--~--~''-'l-'-'------· .... 2931.. . .. 66f1!3-.... ~-----29!1--- 68134{· 2782 61489!: 2832 6~d001 28f2' 648167 2932 66f94t· 2982 68162~ 1 a783 . 61~&o:~ ... _______ 28l3. _ . 63129l ............ HA! .. ~---'-'e:u!J 2933. .661469----~-----.29&3: 68t6tt·

~ ~-~-,-e-.--~~-----,ufós7 2134 632299 a à••· 6481o' 2os4 66~2tt 2984 682t4i . .. _ .21Jt5 ........ !1.fH1: 2835. .632131.. J4.8S• . 648•3• 2915 666&61 2915 68!3121 1

2786 61628,, 2136 632Sit !&$6 649419 2936 66f,41· 2986 68368S· ______________ 2Vt7 .......... 6.1l4al·..... 2837 ...... ..633.l.fl.3 .... ________ a8J7-·-···--64.04tO..... 2937- -······-'6-7-019---- ---20.&7 684161'

a78e 616!2'' 2138 633&83 2éie 649647 29sa 6613(1· 1988 6&438{· ... - ...... --·· .. ..tl89 . 6175119 . . - 28.3 .. 9.~.---'3-38.8.3. .. , ••• ~ . 61-060'1 -· 2939.:-~---c&-&f-GS,:S;.. 298!1 . 6848811

2795 617989 2140 634267' 2800 651063 2940 6680i3 299ó 685361• 2101 .617989 2141 6!:4105 ièti1 . 6J1.16' 2941 66tl49 2991 685368' · 2792 618301: 2142 635011 2802' 651663 2942 66i819 2092 681-13

_______ _ __ 279_3 . 61~1:u._ _ ... ...!.8.4.1.. ____ .U.!!.7t _ lao:s. 65-18~5-------- -1943-- 66-Àitf a9ts 6asoa.t, 2194 6t~o21: 2844 631371 ~a••· 61267t 2044 6688'' aoo4 686'''

-----------··-···~-·····--·· . 21 .. 9! .6193.0~ ........... - -21A5~-----6SU!.t!•..... IlOIJL 6116'1-!'-···· -294-!· ··- - 66J64!' ··20t5•--··· -6869-'4f-2196 610!49· 2846 636619 2806 612,01 2046 61§029 2006 687,333 a7.9.7 &197&5 ......... 2a41 . 63&6s.o ~••' ''3a23 .. 20•' 61o21~.. '''' 6&''•t. 2198 628203 2848 636115' 2808 6l32ós 2948 &15461· aooe '''''1'

·---~---2'l29 ______ 6.11J2.9.!.: .... ~---·--··'··--····· ... 6!'l413~-- .... ····~- ....... 3.43 ...... 2940 67ä841- ·····19f9:.... 687·'''' 28ÖÖ 620203: 2110 637063 29ab 6143Ó3 1980 611340 !Öói ~881-~

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N., ... N A3:

Page 56: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside
Page 57: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

.. N. AJ .

~~1i~ ~ Jtl:;~' ~~=i- ~~:~::::~ ~ ::ïl :~:::; ~::~ ::i~::: ~;:;_c- ~ ~~::~::::~ . . -----~-~- ~ -----~~~~-.35.ii3 ·--~ _ e~8 4lJ 1 . .. !113~ -~-- ~•Ju!.t!5.. ...... ~J,..Qa: ...... :t.ó!llt 3683.~ .. ---ttis..I,J. - . -~'133- •••••~: ..

3554 a6&Jbl 3154 88773' s'a4 oöse~~ 3614 9a•of:s 37ó4 944333 .. 35ÖS 868887 :1~55 188315 36ÖI 906,71 S6J!I 0S.tJ73 310!5 044,1'1'

3556 870135 1 3f56 888507 3656 9Ö7t6! 3616 9165ÖI 31ó6 041341 ~ -"Ö'~- 870127' . 3JJ'7 - _ 88.1915.. ~-- 36-Ö.7 -1Ö-7t.6t- 36J7 - C)Jlt911 1 3707 · -- 941677'

ssöe sto••'· 3~58 &81111 :si~8 ·oó7407 36t8 oaft73 37D8 94167~· ~ ___ -~--3!Ö9- _ S7tasi· ----:tiJ.i ___ . .a.a.ttlt-'----- -ua-• - tja.as.' .. - -- 3619 ~ ~ ,.,,,,. ~ 37öo ••6tat

3515 872Q07 - 3160 889539 3.!b 9Q8673 3660 91.161· 3716 046853: 35l.1 872oli7 :tt6t a.to4B1 _ !6.{1·· tó9at7 3.661 9a•••' 3711: ,.,,,, 3512 8721-7 3162 800787 36tl Oti9297 3662 oa~tlt· 37i2 946871

· ~ ... ~ .. -.. ·------·------~;~-----··-t;t:if~----1-::t-----~~:ttlî··· -·········-~tf:·-··-·····-·:t:~::--····-----:::~--~-- -~--:::i:i···· ---------:;t:~- -~ -:::::r'····· ---- ______ -~ -·-----·· --~---~~----------3.5_1!t ___ ~_87U~L------~MJ ________ ~_9UU._~-~-------'-'ll'------'10.JJt_~ ----··---!US. ... IJi.Oat~ --.S71.J'----94&M,·:.~.

35i6 873591: 3166 19!019 3.Ï6 910~25 3666 IJSQ3P3 3'16 948910 ~-~1?. ".~Zl~'1:_ _ :S.t-61 .t~ltu ... 9 .... _. ..3.6t.f f11i4! . !467 - 034633: 3117' 0.4t,ot· ~sis 874527: 3168 stsöt9 s&ia 9t1ft3 3668 t35873 s1ts tsöótt·

..... ---~·-·~-----~--~ .. -·---· .. -~ ____________ 3M9._-___ lZ_U_V.:__ ____ .ll.6.i. __ .. ~8.9Ull .......... ~ ..... !6Ït-------9-12Jói---------.3.66-t.• ......... OJi~49 -··--···---·-·3-7-!-9"- -··--OI{lt-!1····--·-····· 3525 874st7· 3'?o &04219 ~lio 9124&9 3&7o oaraa1: 3716 ososot·

.. 3521--- _87!Jt9i_ .. !.Î-1-1 .. ---- __ 8_1JtJ.79 -36ft~- --'-Ot2t6t -36'1' 03Ï81t· ~37ti~ 9506!1!• ''*2 S75ss! si72 894915 3~~~· 913281 36'2 osio's 111*· 9S1Sts

. 3523 811.1.11 3173 !95;75 !lt3 tt311! 36!JJ. 911409' 3;13 95t~U. · 3524 8762d7 3i74 89!,31 36~4 ·9136i7 3674 93~461 3714 ''1''' 3525 ~ 816687_:_ 31~7-5 . ''''31- - --:Jt'' os.•t•s- s&?s ~ •sso-1 . 37ts - •st•t•· 3526 877263; !,,, 896067 !lt6 915273 3676 93~817 3,16 953431 352'1 &l7aA:s; __ ---'•'-' st6.40-3~ ·~ 36i'-- 915113 3677 - os.;ttl· 3711: •ss-13-J· ~528 87158?' 3~78 8968!9 ~.,. 9t558! 3618 91.601' !718 953910 3!29 .. 117 i2~l· ... SS79 U'lJ!J. 3t~O 9t6-U-t · 36_,9 -91;.6-öt · S-1!9-- ti•IGS9· s5s5 818547 stao ao7t95 ~~~o 111211 3680 •si&~! 3116 954391 353t· 87_89.3.1:__ !881- --- 89,795 3631 -Ot71t~~ 3681 O!t.&ö-1·- - -S1St· 0S,Ó61" ss:!a a?oao!· siea 891515 3t~2 tt7137 3682 •s•18S 3132 9s,aso

.. 353!-.81.9 .. UJ, ________ J._f83.---- .. 8988.$.9-. -3683 ----9!;76-tJ--- 368-3.. ~ O:Ji.jl'- · --37$3 ····~· 05!Jtt1J 3534 8105!5' s,e4 898563 3f•• 'i'''3 3684· ''~''' 3734· ,,,.,, S-5!5 .&a.nst.s:.. -~ ..... .sJBs:. .8.99!;19 st--' 9s,.ftls 3685 os;t4-l' s;s!s, -os6•••i 3536 8807o7 3~86 890123 !686 Ot8617 3686 os;&;:s 3''' ''4''1·

- !.5-3.2- . - 8l1aAl: -- --3f.l2-~----'-Q-00-1Sr--------36,'1-------·.9i.8tJ.It- ·· -3687 · ···· 9J; 6-'J 3'3'1- ···- 4J'15!-9··· -~ 3538 8816t9· 3;aa ooót43 31~8 ''''41 3618 os?sao 3738 os;8a7·

--~- -· .. !!139. ______ J~8.i.1!1-:----~--~-1t.8L. ---'.Q-Q.lf1--·· 36~0---~-9i964l' .. -----·3&19- ---.CJ-!8-8-40-~ J-;!9-L-----958"Ó0t· 354Ö 881459 3100 901~43 3640 919833 3600 930411' 3,40 958091· s541 ail29tJ.~._ sttt - •ot443 s ... 1 o2o'•' s••1 03~'~'' 3'141 ossalt· 354a &8340t: 3192 9ot587 3642 021oe1 s6oa 93•''' 314a OS0!4'1' 3543 . 81114'}1:. - 3t93 9.()~067 36.J.,- 9!1.207 3693 945!!11 -- .. 3143 05914'1' !544 8837ö7 St94 902,39 36A4 021201 3694 94ij?41' 3744 959363 3545 814511: .319!5 ~··~ 901931~ -3641 021045 ~ 3-605. t4Ö14!J' 3'145 950611: 3546 8851,1· 3~96 OQ!I67 36~6 92232t 3606 945,45, !746 96ÖB63. 3.!47 88!i5.3.87 !197 OÖS61t 3647 922129 1607 048084 1'14? 96ó8!!J1· 3548 &85387 11te oàsP39 36~8 922177 36oa ••isö1· 3148 t61oo1·

.. :!t!l4.o. _ ~ s.B&töi.:____ . 3199 . ~ -•os.rl!o. ~••• •a a a'' 3699. 94-to'' 3749 o•t '':s· 3555 886443; !800 904089 36~0 923817 37ÖO 941241' 3'155 062313:

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Page 58: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

A.l. ..J" . A.3. ..L ..•.. N.. .AJ N !•: N AS

3 15i 9 62363- !I D 1 08~817 --~~~1_- ____ 1Jtg_14t1 ... ~--·-·· . __ :U_ÖL ... 1tt.!.g.8.tz ____ ··-· ._:tU1-~.-1ö443A~-----~----·--··-~----·-··-··-----~---~--- -~~---~-------~-,,2-- - -olas3J: -- --- !ilfi---~-,~~o o1 , e tz 1 q a 168 t t~o a 2· 1 o ~~ 64 !I· :~ ''2 · 19.4

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........... ----------------------.. ---~ .. --~--.. --... -----------------l~;~ --~ :::~~l--------~-;:~l-~--:;:~l:r-: --- -------~-~~l---t:tl:;-~- ---~}~!~ -i~·:t~tl-------~:-=~ ---i~:t::;·-3?55 964Qt~ 3805 082835 ~,,,. 1ÖÖ241;5 395!1 1.0 ..... , 39!1, 104191'' ·:rt,--6 t642t1: si~o 6 •••o:~s 1&86 10 oa•s. 1 39o, 10113•• so!l& 1ó41:ss., 3751 964619 .. 3101 .. ·~~0~!5 !J8Î'?'_ 1Ó~li!l_, 39~7 102~511 .. - _39Jt. 11)4164!1~-

~----~---- -~--~--~~-- - -3-,,-ir-·-- "''4'~''~ ~- --:s-.oe--~-,.4131 sas8 1oo3e'' 3to8 1oas.,,,. 3918 104119ot

.... _" _____ .. ---}l:~ . :::!:~-- --·ltï-~----ll~:;~ . ~~:-:--: --ili~:g:-. .. ~:~:-. . i~:t::t ~::~- i~::::::-- - ~'-~-1 9'-~4.~.1' .111.1_ 981119 Jl-1' . 1.0Ö.,_4.~.1 391.1.. 10Jil61: .. 19.61 -104:Jtt' - 3162 06&9a3: 3112 tatoo3- 31611: 1aö!5113 3912 1oti149 3962 104463,.

3163 . . 9t7a & 1· . :n__13 986413 . .. .. JJA.3c. ... .11lÓ'.1...1.3c ______ !J.i:s, ____ J • .C.tÏ4J.2' _______ 3_9.6!:.--~-1-Ö.44hl' ····---- --- ------ ---------·--·---·---------i-f64~----,,76i)-- ----3&14~-----.-.-,b si ____ ----- 3$64 tQ ö ~~ o t :so ~.. 1 ó ~~~ti 1 3 964 1 Q4 48tt ·

------------·--------·-----.3.!.6.!. _____ __9_fuft~ ____ ___!_W ____ ....:.tffi.U ______ ~--11-U'. --U-O.A2.8J. .. _~------!.t-t.!. ... ~--1-4l6"A'-'------~3-'U~----U41-1•1-.... ·-----------· 37 66 968!StH: 3816 98139!5 3lt6 1Ö Ö614t 39{6 10t6'1Bt 3966 1Ó46lU 1

31&1 . ..9.t$5.31:. ... _. ______ U11. . ~-- 91,7lU. . 14~1· 1QQ1t'12 . 3.9~? - :&.c$-tt!J' - 3061' 1~463!1

----~-------·--··---------·--1~!: --· ---~-::;:;_~~-----:~t~-----~:.::~;:_ · · -··-·········-~l!:~-----1:!.~~:t ___________ ~:l;• ··· · .... i ::i:l~. ~-···---- ..... c~.::;~ ....... i~~ ~;.i:: ...... . 3775 ''D11s· s1ao oea8ss 3tto 1qgooQ' 3tao 10I?98t - '''~ 1o .. ,,,r. _:tt71 9_7.17~_3_:_____ .3 .. &21. .. _____ f8_~96.3 ____ .-- .3$7-1-- .. 1D.O-'IL0t -- .. 3911 . -1Dif41S:. __ .. - .. 39'1!:.. 104B17l· 317~ 070763 3122 090395 38;2 10Ö9l33 391! 1018613 3912 1048311

. 31'7..3 911531: !123 09ó3.95 Jf-tJ. 1ÓQ9469 3123 .. 101--IÖB 3013 1348JSI' 3174 9721Ö7 3114 990305· 3$14 10tOG93 3914 1010,61' 39?4 104913!· 3775 .... ~7attLZ.. ~- .312.5~ 9.91.403. !.8l!: - -1tU.0.660___ .. ~ :!91!1 10$Ö0-60' !91B 10,.0131

-- -3116- 9721'9 :sla& oos.7B7 se;,. 1010''' 391& 1osre••, s91~ 1o4o9ms. _37'l7 .. ~.725!tJ.L .!U1 . - 9.91!1-5- -... $!+7· -1Ót131'1-- .... 3011 .. 10Jtl60 -3911 1Ö50!I9!· -3718 9728d3: !j28 991!07 sets 1Qi2t81 3018 103!389 3978 1ó!t071: .32.19 .973.5~7. . !12.9 9.91987 !l?'.· 10l2ll1 !OIO 103t4fft · 39'! 1êS!S!II' !?85 073931: !830 003!15 1890 101232!! 3930 103~91!1' 398Q 1f11'-'

. ..:5?~1 ... --· ~7~2.7' !lil___ 993!1!5_ 3881,.: .1Q.1!1J9.: ... -- ~39.31 .... -103~110 -391t -1051561 3782 974105'. 3832 00370? 3882' 1013477 39!2 1033189' 3912' 1051899·

__ --~1!3 ___ '-~~_7_0_!' _____ . __ 3l3.3 ___ 9._941!!5. _______ -~l·3- .lQ-~36_60 -- _3933· ~--103JOÖ9· 3983- 1~!JI8f9: 3?84 974987 3&34· 991!11 1$14 1014129 391~ 1034213 3084 1083043

__ _3!~'- _ . «>.I~,-~3.: _ ... lB.:S.! .......... Jt9 .. !ll!........ lf$!t . 1Q~460!I 303-5 103~~~~; 3985 1053335. 3?86 91631~- 31.36 995&75· 3$et 101!5837 3936 103443'' 3986 1~5443!1 1

__ __ _ _ ~---- _. ____ ---~~ ____ _ 31_~1- . __ !Z~~ f" 1:_ _____ .. ..J.I3J' ________ 9..lJU1-·~--- _.3 f'-'~-- --1 q 15 á-3.7.'- ~---U-3"-------10-3-~~-~- . ------39-t-1.:-~-!-9-6-47-0t-- .. -3788 976691 3838 996347 3888 1D15t4i· 3938 103.$0~- 3088 1054963

__ .......... ~---3-'Vlt .. ~ ..... -~1129 .. 1.: ... ----38!19 __ ~----'·'-'-"-~-~--------3ttt----1Q,.S-1'13-----·-----~·-~·lf-39----10JUI-I'---·-----S~t-!··-·- -··1tlll-39• 3'195 97?7~3: 31~0 996347 3~00 1Öi6J8t 3940 103it13 3990 1ÖJ671J 3'19i 9({72:5: 3141 .996..123 -~·-· 3801' 1iit7l!l1 3041: 103J6fJ. 3091 1Ö56Jtl

.. 379a 911861' 3142 997~03 ~soa· 1Îh,7301 3942 :1.03,01'1 3992· 1011011

~~l~· :-~g~~r-- ---~:!:-- ·--· :-:i::: :::: i~t:::; -- --:::: -·· i::i:ä;: ~==~ ig:;:g~ __ . ~37~9!5_ 07964..3-'- 384!1 .... 9.98!13 -~leiS .. -1Ó18387 . 394!5 -103fOI!JL 399!J · 1~5863lJ'

3196 079787 38~6 990319 3806 1018,89 39,6 103868!1' 3996 1058631 :,H1.7'. ?,0483 31A7 99.9329 3897. 1tÜ,9!JOO 3047 101~001: 399'1 1058913: 3798 980843: 3848 999665 3198 10t912t 3948 103918!1 3908 1059403' :s1oo 9&oaA3 s••• .. 1..o.oo!.l5 salt. 10203.97_ _,,_., .tost&&; stt9~ . 1oso•ós~ 38ÖÖ -cj-&fi•-,--- - ·ia!lo--. 1ÓOOB65 39ÖO 1Ó20307 39!10 104Ö34!: 4tJÖÖ 1059!54,.

Page 59: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

.. ~:5 .. . A.3. .N

4ÖÖi 1060411· _41ft ___ 1_0_8D~~~ _ 4ii1 11003&9. 4111 1t2ä485 42tii----11Aii!t!1: --·---·-·-·-··---·-··--·~ --·-------------------·•aoa-· To&o7«r,, •••a 1oao.,,1 •\22 11oo•2• 4112 1tto62e· •ao2 tt•itas

4003 _1DJ1tö7_ 4è!t:S __ 1081115 .4~g3 __ Ü.Ö0629 4~JI3_ -1{2iUt· ---4243----U.4i6tl-------·· ----------------·-- ·-iol4 to&s.sa7 •••• 1oats'1 4~g4 ü.as.e13 41!14 1f2iolt· 4to4 t1~t~)4t· '.40.05 1D52ó~; 46!1' 1D.81J,.1.. . 410!; 11Ö1!19 •tiS 1tiÎ3ót· 420!B 1l4i44!1 4ÖÖ6 1062427 4fl6 1081911 •iö6 11Ó211!J 4186 1tài3ö9· 4206 1143061

________ ----------- ·-- -·-·--- ·- 401'1 --- iO 62427' _4!11 10821.91.- -•&!1· -110-2181 4111 · 1tlS2t3: 4207 1143Qitl 4ÓÖ8 1062619 4i58 1Ó81839 .,-, 11Q2611 4118 1!2'''' 42Ö8 1i4306t

··-····-···--····-··-····MUit __ .tDJ~ólt.:. . .. 4159. --10.8!!!19. .As,}o. .... 11037At.:.- ···---4119 tta;._,,.__ -·420-9 ----1143,41· •oio 106Stt' •o•o 1oass19 4jio 11ó4t6s 4160 til~,,~ •aïö 1144071'

t~îl -!~ 2::~:·. :::~ i~-t::i~ :tti ii3:!~~ :i:i iti;J:i. ::il it:=~~~: ____ ------·-·-- --~--- _________ . ____ .4013 __ .1Jt64_9_2_5_~-- 4jj.3. __ .. ___ .t.Q 8t!!!L . - .. --•iÏl-- - 1-1041!1- - ----4-143- --t.ttflt-9-· -4!-·13~-----1!4H':J:

4Ót4 1066147· 4f64 1081047 4fi4 11QS461 41&4 1~1-'-41· 4214 11464~~: ··----~ .... ------·--· -- .. _______ ..... - ___ ·- . 4.ii!L _ .. i.OJ.!t.!Z '_ ___ _j_o_l!J __ ... 1D.ll8l.!t__ __ __4 til'- ....... 1t.0.5.Ut~--------··-4-1.4!L- ----lt.tlt'3~----- .. --421·5-·--· -1l:46-4!JI;

4ÖI6 1D6631J~ 4i66 1d86~01 4fi6 tifi637S 4i66 1tl~961 4216 1!4664. .. Att11 .1D.6t_l9.1.: 4t.J1 1~!.6..8.15~. 44-l1 . 11Q666.t -4ilt7 ····· ·1-tttt6J· 421' · 11412-&f'

--···-- ·---···-- --~--------:11: ______ ~-t~ft~. ______ lUt ____ _tl_:Ii: ~--- ___ _ ___ !.it1 ............ ifl~n: __ .. ___ ........ !i.:: ___ ti:Iil; ·-···--·----·· JUt-...... ,ii:-~::;.:. 4Ö25 1d67!75i •i'o 108812? •ito 11Ö'''' 41~o t.itA6t:J. 42tij 11Aft11·

__ 4o.2.t... __ tole 61_~_ .. _ .... 4Jt~.t ...... t..n8.ll2l.... . •i u ---.1lo.BI!3 ·- -'111· 1s.aoo,,: ······•at1-· 11488'1?' . 4i22 1~69267' 4Ö72 1088,43 4ii2 1~Ö9469 4172 1ll~260 4212 1140161 4023 1D.6P2AZ 4f13 1Q89415; 4fè3 1~Q966t 41 '3 1!;1!613: 4223 1!.4016, 4024 1069Sö7 4.74 1081895· 4!14 1109661 41?4 1130111 4214 1l40!61 402! .iDZIUil'_ 4_0,75 --1..010.063. -4ii.J - 1Î$,0137 4t1!J- 1~1Ött~· 4215' -11496!Jt 4026 1D71Ö4J. 4ó76 1ó903'S 4116 11106.1 4176 1~3Ö301 4116 1i!ÓS99

..... 4Ö.2i. ..10 7UJi _____ 4j17_ --10909!11: Ai.i7.- -11t061l--- --411'1 1~361,1' - - 4211 11~1131 4Q28 10712''' 4ij7S 109128? •iil 1t10,11 4118 1iSl540• 4218 111{331· 40:29 10 7113Y 4119 111912.17 4.110 1111i.9 . 41 ,,. 1tltt.S3-: · 4219 11!1261 4Ö3o 1072219 •aso 1091281 •ilo 1111121 4t8o 11SÎ111' •asö 1151987

... •-9.31- __ to12at.o: •1•1.- -s.~t91863 4iss. -. -1~~24-41-- .. .... 4t8t 1~~-~~~--- 42-s~>- -1t't's'· •o32 1D722t9 •eaa 1aoa~a' 4tst· 11121at •1•2 1s.:u11'r 4232 11"sas1

----49.3.1----1.0.?U.ö1' ___ ...... ~-'iJ1--·---~1-1J.f2'-2'1-- ... - 4j·3-··· ·11-ï-3-12!J···-·- -----41;8-3- ········tt!Î8iJ' -·4-28~· ·- -t1~Sf6~'--4Q34 1073323: 4j84 1092911 4tS4 1114DQ5 4184 1~SAOI1· 4234 1154301

-··---!~:-- t~il:ir--· ---- :1::: ·i1::::~·- " :tt:~ ttt:::: -:i:: lt:;:;:_ :~=~ · it~::t~: · .......... 4.Q.3" UL1.l4?...5: .. -------.J.fU--·--10-'_.!..0-J:__~-----41t~~---1-~~4111----~----·4!-l'--·---1~3-f6i'"-···----·--42:J,1----11551h"···

4g38 1071955· 4f88 1094887 ,,,a. 1115211 4118· 11!,6CI 4238 115!J81t _ 4Q.3~- .. -1.01.~_il.l._. ______ j.-tJ.t_ _____ 1Q95.!61 .. --------'-~~~--111-513-3------ -··· 4-t-89:. ---1tf60J,:. -423!·---1!S!J81t'-

4045 to7S20t: 4fi9o 109!1&1' 4i•o 11t.5073 41tó 1$16813 42-o 1tS9t7s·

:ï:l-- ·t~~~~~r- · :t:; tg::::~ :~:; ii:::g; :i:r ii:::::: :::i :t:;~;t ~0 43 _1.D7~l27~ "· -~-012.3. .1.Q9_6~19. . Al43 1-116133 4t13 -- 1SAI,IU7 - 42-3 1t.S1961 · 4544 1D77tt.5' •IP4 1ó919L1 4i~4 1i17029 4104 1t..i077 4244 1158279

:~:: i~f~::;-: - :t:: -i~:~:i;- :t!:' itt~:~~ :i:: -1i:t~:r- --:::: ···· ii::::l 4047 107f!li7. 4jt7 1091301 4t'7 11!.856!5 4t.01 1~10llt 4241 1!S918Î' 4548 td785Ss 4lt8 1ooao13 4tia t.tt.&lös 4toe 1i1~'33 4248 115064~'

.. .4040. .... J.0794t.9_ _4j9.9. ___ 1091313: 4t4t- . -1119815 410-9 · 1t!t?33: ·- 42'9 U,6Ó~I1' 4Ö5Ö 10l01'1S: 4fOO 1099813 4t!JO 1t20481 4200 1145069· 4!!1Ö 1160815'

Page 60: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

.. -~---~~--- __ --·-·------· .... A.!c N. .... A3 Al- -~-- ---A-1: · · H A&

__________________________________________ --~.2~1 .... _u._~_m.t1!~--- ____ 4J_a_1: __ 1i !251~~ ___ _j_~~-1~ _ .. 1J.la44'l: ...... ___ .. t1cfi1- _ .tais1ttt.: _______ .ül!.:.---t-a4411'-t' --4252 116!279 4!02 1182999 43,2 12Q2~0' 4402 1t2J!~!I 4412 11!44793.

------~-------------------------------~-------------------------- ---l--i;~----1iti·I~~~--- -------1l~-~---ti:H~} ------::1~- ---t:-~~:i~--- -- ::l~ i:~-~:~- -------:::!- il::::r At~'-

1trt

22_. __

96:t ~IQ !I 1S.SS~47 4~f! 120~079 446!5. 1.l2f629 44!15 124!19SJ;

42!56 • ~~ 4!06 1i84019 ··~6 12~4469 4406 1221865 4416 114,21'' 42!5? U.6S367. ---~~lo'! _ t.!'~'11__ 4Jf~--- 12g~_829 __ 44Ö7 121SIÓ!I·_ ... 4417 __ 12469!17 ..

- -4!58 - 1.f61615~- 4108 U.841199 4$18 120!5311 44Ö8 1126041' 4488 1141511

--- -t~~l- -ti::-::r :-:~r--tt:::~; --- -- :~::-- t~~;:l~ ::t-: -i::;,:t- :-::~- --i::::;;: -~2!ti· U. 6!o 9S• .~J-~.1 ... 1i8.190.3.. 4Ji1 12.Ö~.4lS 44!1 U2illl 44ji 11488~1 4262 ü.6:7·3· 4~12 118&119 4162 12§6111 44~2 U!lf910 4462 U!4028t

-----~---~~--------~~-~---------~-~··------~~~3 ______ 1.1 .. ~.:..1.~--'~-~--~~---!ll3..:. _____ JJ .. S .. 6_,,_1 __________ .fl6 .. !c ___ 1to11:!.3 ... _______ 4_4t3. ........ 1!U!!7.~-- ..... 44j!-:. .... s,.tA0281.'

------------~---------------------~ --:m~- u~m----- i!H-tt:mf- ---m~m-- ... -::u -rnmt- -~m-- ~m:;:-. .42-h1 ...... 11.6J.~.t9 .................... -'-f1'·-----·-..U.-4.2'.1.4.1.. .... _____ . ·-· 4W."- 1tQ 81.~ 4 4-Ï 1 ........ , ... ~2!1-' .. ,,. 12JQ&!-!'

4268 116Bae1 •111 1!&8231 ,,.. 12o&t8t 44!& 111187! 4468 tt5o?t3 --~----···-------··--- ~-~-------------~-----~•~-~~-----Ulti!.3..:. ________ 4.!1_9 ____ 11JUli: ... _. ____ ._4tu ____ ~ujH.H ________ 4.4.~.9.----~1-t.JÖSII.:- ,~.A.l9---11-h-tt7:~--- -·--··· .....

.. ______ :.~;~---····-- ... tiJ;:_:r _________ :ll~~--tt.:::~~ ____ ... :l;t ___ l:f:J5:__ _____ :_::t_ i::I;i~.:-· .:tit __ .. i:::::t: 4272 1169535 4.22 1189?19 ·~t2 121Ó821 4422 1tll1'' 4412 1182913

... 4273. 11t982.3 4523 1:11191-1 41-'3· 1111661 . 4413- 11-'•''' . 4413- 128:!77'' 42?4 1170831 ·~14 1~9Ói03 •lf4 12!2629 4414 12~•''' •• ,, 1254089

·--·--·--·········4.2'l5 ___ .1111_43 .. 1:_ . _______ 4JJJ ... .tt-IU79 ............. 4-Jf..!· ---11!.26.!.9..... _ _._4J! - .... -113.~0 At· 44'15 .. -1-1!5461t .. . 4276 11;179~: 4!26 1192047 43?6: 1212.41 4416 1·$~401: 441t 1254610,

_ __4t'l'l__ 11.216 _ - - . __ fJ.27 ·-11i2Q41 -- - 4$!.1 ---12-1.34169__ -4411 - -11SS7J!. 44'1'1 11151001. 42?8 1112111 4é2B 1101311 4$!8 121~101 4428 1ISJ0'1 4418 12581'11' 4279 11727,1:. 4!29 .1193.1-71 . 43'' 12Ï4215 44-29 _,J-14.617 . 44'10 12!515'1'1' 42ao 111!1~3 •t3o 1tos'1o3 •slo 1214215 44~o 1•sieo1 4485 125,&28'

.. 42e .. t 11?~?.--!.t: _______ 4831 ...... 1194119 ... _ --·~•1 1ts..u6o ________ ,4.Jt 11sia•1· -'-41-i 1•15t3o-s: 4282 11~:ó1n: 4832: 1~946'11 4312 121!431 4432 1,s;e's. 4412 U!5617~· _

_____ 4_2~:~ ____ ..11_-'~~z--~----~~-ua __ jt9 .. Sl1.o1 ..... _. ----·~•-&·------1-2.1!."-s'----- ....... 44:u --·· s.ast'•' 44a:s 12'''''

_:~.: ... _.ti_~~:;_~~---- ___ .. :m ___ ~ii::.:tt ____ .:Jt:: i:i::.:i.· ____ :::;. i::~i:i. . ::::. t:::::~: 4286 11719!9 •• ,, 11'''19 43~6 11i73Ö9 4436 1'3f961· 4416 115869?'

-------------------~--------------------------4.2~'1_ ____ H._!,.9_1.9 __ ~~----!.ll'l.. .... _ut.6.3.!1 ............. 4U7-'----1.21-'-54----------4-4-1.1:.. .... uJ34Al.:...-... ·-·----··--«-1-7 .. .___--1H-169-'1~----4288 11719C3: 4~38 1iO?n71 4S.8 12i?933 4418 1t3i969• 4488 1158889

. 4.2.8.9 ....... 11.l..'Z.2S_1.~---~-41ll .. --U.f'll-'..9- ..... -·-·· .... 43-IO· 1218109---··~ ---4•39----1-13it6-t -··-------44-U---~1!-IO~os· 4299 1!7786!- 484-o 1~07163 4Soo 12~8701 44'0 1J3~1'1' 440ij 116g35-·

.. 429.1~ U.7fh,1: •••1· 1!91AJ7. ~3-91 .. 11!8'19'7 444! 1110133 4401 126Q'1!0 4292 1178439 4!42 1108127 4302 1219!51 4442 1145289' 4402 1160830

..... 42.93. .117?6$7' .. 48t::s .J.~ol.,27 -•3-t.s: -- -12S.944!L 4443 -1t4i'•' 44t::s~ ~t6iJ!tl: 4294 1179,t9' 4844 1100063 4~04 12204!3 4444 114061!' 4404 11611''

__ 4.2~' . 11795!1.9_·__ _484.!5. .11996!9. __ 4-J.ct!' 12aot:s::s -444! -- tt4il-ts~ 4495 -1t6tt••· 4296 11799d3 4~46 120~839 43~6· 1211D~1 4446 124i4ó1· 4406 12613'9 .4!91.. 11-:fU:U' 4!141 110,113- 43t'1 112112!J 4441 124i40S.: 440' 116~831' 4298 111086!: 4!48 1201213 4308· 1222061 4448 124i!69' 4498 1263167 ~299 ___ .U.tlZ95~ __ 4!t.9 12tU.f2'l 4-St• 1212.161 4440 114310:1: 4409· 1263911: 43oo 1181463: 481o 1to2•4, •4öo 1222111 44!o 11•~•,, •soo 126441!;

Page 61: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

..... JL 43. .. _.À.! .... . .... N A3:

Page 62: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

_ ... 4.3· tL ~---------A3. ____ N_: A! .... N... - .• 3. . -N . AS'

---···-··· .. ········-··------------~--------·------~-?~1. ---· 1171 'I' 9. . ....... !tll.._~l39_i8.1_~--------•jii·..... 1415t!J.'---------~-~-'.Ö.1 ____ .14.1i311~.... . - 49!i .-1,4.!.l9Jt ________ ,, ... .

-------------·----- ---~ ______________________ :~;~ ____ f:~1::~: _________ ::~~ ___ t~_mJ:. _ :tJI_ t:t:x~.: _______ ::t~ _____ i::l~~= ___ . _ : .. ::: .t::i::t ___ _ 4754 tS732~9 4104 1394187 4tS4 1417467 4904 143i2ó9 4984 146t4t1'

i~;: i~~lil~·-- ,,, ···::~: 1~:::~: ::i:. i:t~::~- ' ::t: t:l:t:i ·:::: i::::::\ __ ~------------- ---~--- .. _______ 4.Z~Z __ .11l71l~-1 ____ ~--U.tL'L ___ 139,939_ _ 4if; __ 1.4î.S.&t.t __ ...... 4.9Ö7.. 144QOIS,: - .... 49J_,. -1•6.1~-0l-··- -

4158 1~71o33. 4èoe 139622? 48SI 1418889 4958 1440177 4018 14630a'· ------·----------------------· --------------------- --··----~-~~~----- --t~~-;~~r--------:-J~~- i~-:~-~;;--------!:-:-~-----t:t:-:-;:------·-····----::~: ·i::t::i--------:::~----i:::::r-~

.. -~Z~.t 1~?..~2~1 . 4_{11 139lD 0.3. üA1:. 14t.OJ?! 4111 1A4Ï6'-1 .416l--- .14644.'17 4762 1~7&&71· ••12 1391&4? 4è~a t4ao''' 49i2 14417~~ •o•i t4••••s:

··----·-~-----!~!-}--i:~~1~t:------::~:-·- ... i:::::~-~--------~:~----·i::::;:-·-······ -·::~:··-i::;;-::~--~----~-:::!-- ··-i:::::~· _ ----~---- _ ______ ___ . _ ~ ___ __ ______ . _41_6_L ___ tltl.!t~3__:__ ~ ___ 4h.J. _ .. 1.39!!.0.3. ~-----«&1.:...__ -1*11 41t · 401!5...1A.43-0U.:. .. _______ ,494J -- 1-4 6 6 OQ l· ...

4'766 1~785~3. 4116 1390899 ··~· 14!2t~3 49Ï6 144f4~5: 496~ 1466673' ~'l.6.1_ 11H~.Sit3 4111.~ .... 1JUU!L...... . 4.JA7 142231!!1 491'7 1•41Dól 496'7 146661,. 4168 1~716!5: 4118 140Q9'5 •t-a 14!2609 49t8 144ii6S 4968 14670!!IP·

----------·-------~-1-~l ____ 1_~[~_611· _ ...... _4.li.i ____ 1..4.0i.~ti'L ______ .48&9__ 1.423299 __ 4919 .. 1441261· 496.9 . 146771-t· .. 4'775 1~8$631. 4110 1401861 4Î;o 14!3827 4910 1449113 49'15 146140!.· 4.!71 1~80919 . JIJ1~-----1~Q.tl_l2.. --· .4l.h ...... 142312,_ _____ .. 4.t.U~ . 1A46Ut· 41?1,· -- 1461-111L __ _ 4'71'2 1~1S3!!Jt· 4122 1402905 4S;2 1424t1!!J 4922 144661, 4972· 1468881: .• 7'73 tiU.18!1: 4113- 14~2tl! 41"''' 148431! .40-~3- 144;-0t1· 491:r t-4606!J:Jr 4'714 11581!3!1.1: 4124 1403499 4SÏ4 142!!1243 4924 144Î011 49'14 14'1Q20t:

---·---- ·--·-- ·-··----·--·--··------~ ___ -~-------"'- 1. 1.. _11llll.1._. ____ tUJ_ ___ -1 .. 4G-183.5~. 4.!11.: .. -1A-1.5.S1.1. ----~---4.12~--- -14-4181 7~ -- ...... 40.7!5-' - -1-19-1-G!.' 4116 1~88647 4126 140!215 4é;6 14è5163 4926 144817, 4916 14?Ó20t:

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. ~4 .. 76.1 .... -~11 __ t,J.ll - -- - .!1:51~ . 1.40 119S -- Al,-1.:.-----1'-1-SJ:n. -- -- .. 4931 -· - 14,j.!'-O--- -- ----40-.1-!·- -14114-!11 ~?ea t5S6d~' 4132 1407319 4812 1428819 4932 145o&J7' 4912 14731!3

_ 418.3 ....... t3l~-Ó-''L ~ - .. 4.&33. 14-0-7915. -- -------4&-t-S----· ~1419-1!11 . - 4933: 1415Ï160- - --~4-913 141:JPt$ 4784 138624.7' 4134 1401347 4éÁ4 1429615 4934 1451441· 4084 14711DJ

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------~--··--··-·

·~· ···--····- ·- - ·- ·--·· •.. ····· ··-·······-- -···--·······-··------···-·-··--·-······-······--···-····-·-·-·· -·-···-··-···-···-····--· -····--··-·-···-·--·· .. ·········-· -·······-···· -··- ·-

Page 65: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

......... _ ... Al·. 41 .

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''"~ 1'1ns.:s1: s*•3 tl•J&31 ,,as .tl56t .. 1a ______ .!.6!1_3 .... t.,aö4~·-·- .. .. ''·-~-3-:.. --1'"·.··"'-"-.· __ ·.· .. :.. ...... -----~-----i-5t4 tf1oa53·. --,-;-;.~----1,;,-àif ·· se54 t1s?to• 5614 111i,.,; !!1154 tlo4!H,1· ''o' 1!11• '~':__ _ _,fl5 . 1134933 . sAös t1)7••~- '·'-~' :s.'ai''' ..•. ,;s. . taoaaa~ 55Ö6 1'11!261· 5!56 1'135269 16Q6 t1!90Q5 56!6 1'11Î1Î1 !51Q6 11g6Ó63'

.... !SÖ1 ....... 1:11J .. a.Ll:. ______ fJ.51 ____ ~1'-!5.61.2 !.'0-''~ ---11~9-tC-5 ........ 5.61'1---·. 1'1-lf71S J1tJ-'1-~--1-IGtti!J-'

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........ -~···· . -------~5i3... ..... .1Z1~~.i.~~S _________ S_l6:! ___ .... 11!90.6.1 .......... !!Ï~----1-2,!.19!7. !S663 .. ·-~'&i4.__,: ______ ~!J"'l:J ~--1-lfllO:I: ----- 5514 171!1!65' !1Î64 1731061 ~6{4 1162341 !1664 178!i630 1114 1800063 . .. ..... -~·-······-····-··-··-· ''..u ...... t!~1,_J.ii5.' ...... ~.. ttJ_5 .... 11:3.16..31 ........ -~----'A.is~---·-.t''2'"' ·-''-'• . -~~•6-t.it~----·----· !J"'fs· 1. aö 0111 ·

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----~--...!.si.9--.. --i.ltJa,,,__ . ·--~~U-----1-74-ó.l-4-'----··-·······''·i•- 1"16424' -166-t--- --t·7-ta7•-a· . 5119 111t5tt· 11525 1116973. ~~7o 114t&69 86io· 1;64!17 567o 111oso1· !1115 111taoe·

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A3

Page 67: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside
Page 68: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

. A~.: ....... __ N ... A3.-.. . .... N A3 . N. . A!.~ .N

625i 20 70 1·9· 6Î01 2Ó 05!50 7 63ê:t.· ~~2011.1. _ 64Ö.1 ...... 1t.t!OJJ.~.-- _6~1L .. 2t?öttl ··---------···-······-···~····-·-···· .... -· -·· '6252~---- 2rffèfji3~---------6iö2~---2-o9ia'i·5····· .. ... 6 iii··· 212 0 2 a 3 6 4 6 2 ~~ 4i4 6!1 6 412. . 21 "14 3J.

6253 ... 1o~o 101~-- _,_to.~--~~-q!ftiJ_, _ 6li3·__ 2:t.21.t4.'t_ _ ••-ä:s . JU .. f9.1o_,, .64.Js·__ 2t7aaos. ---···-· ------ 6254 ao7rl3B3. 6804 2ot&st5 6Sê4· 2t22S87 6454 2!.4'3''' 6414 2t,aaas

.. ~e_,, ~o1.2~a~. •tos 2óo1J47 •~!s 2122''t 6455 a1•;.l4J ,.,, 217aate 6256 2072383: 6106 2091!0? 6386 2123595 6456 2,4,649 6416 2t7316t·

.. 62~., zozsa''' 6JQ1___ -~tilt69t} 6~·7· 21241175 646.7 2t4f649 6417. .2t73!U3. 6258 2073775; 6808 2Ö99035 6S~8 21.453~ 6458 '2148119 6418 2174141

-···-··--···---------'-2!39 ao.T!u._,:_ .6!09 __ 2.1002,9 ........ ···-----'*~' 2tt4S3.t .. - ... 6459: .. aî•l7os 64!1o 2t74t6t' 6265 2074311: 6Sl0 2~óó836 6360 212491!5 6410 2115049 646Ö 2114961

.... 62.61. .207!!311~. 6.!1.1: 2ltl.OI3!L. .. A$.61 2125.!.3.9 6411 ~~'i6.á!· .6.461 2t,6l!.J 6262 20~:':: 681.2 2101027 6$-2 212:!:39 . 64Ï2 ·~·fi:'~':___ 6462" 21,6833

------~~····--··~ ··----~-~-----~--~---~---- ··~··· -:::!-- ~~-~7ii,-,~----~· ~-:~i!----}i~ i:~:- . -~--~····-~···:::~-~--:f:6:~~~------~-:!t:. ~:;:~::~ . ---:!!~ :t;;::::~--.. -~ ---~-~---~-----------------·-------~--~..!..6.L_ao76847'~-----·..i.W_. ...lltl.U99 ________ 6l6~---~.2.1.!'l.1!.t. __________ 64i!t __ ·21112J! ---~--------64-6$'---1-1'-'-B''·'· - -- 6t66 207'18!5' 6Î16 21ÓSl15 6S66 2128347 64{6 2!fi4'3 6466 2t,8bóO

-·-- ~.62.6.1.. tol,t.4.3, ____ ... .6.W ..... iliJ.O.Ii.·... ,,,,_ 2tt.834.l 64î'1 2'-ti33'1 644'1 . 2t,l41t· · 6268 20781~3: 6!18 21046'15 63&8 2128587 64iS 2!8!'169 6468 2t,A82J• ' . . 79 ' . . . .. ' . «, ,, . • 9. '. . "' . -, . • . .. . ~----------~---·-·····---··-----t-l,~~---~----lla-(ff,.;~----------ttl~-~~-it~:.l:-·~---··:ll~·~·--til9·tit~-~~- -::;~- ·- ··~il!:&: ···~-···::#&----tt:~=~f·····~ ..

. 62 .. l1 ... 21L~122~~----~--1tS.21. __ ~_2.1.ÓJ~8l5_~---~-- 6!t1: . .. Ul.0.14t__ JAJ~L 2~,30!1 64;1. 218033-'-' -6272 2080387 6122 21~630? 63'2 2130939 6422 218624! 64'2 218Ó48!' ~2-~3 êQ fQ $61 6!23. 210 681.1 63;! 2131305 6.423 2!1AI4t 64'13 2!8S.SS7· 6274 20813-7 6824 210,211 6-t4 2132379 6414 ~~'~''' 6474 2182681•

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. ~-· .6277 2D I:JóJ15• 6S-i'1 ~ 210 l819 6it-' -· 21335!!J .64t1. 2l,t219 44'11· - 2!84009 6278 20841,9 6!28 2109ó91· 6~78 2133987 6428 2t,~289 6478 211468i· . 627.9 . 2014.1.19 ~6i29 210088.9 ttsjo 2134315 6410- .. cltttioót 64'9 2t8.6St· 6285 20844!9 6830 2110435 6380 2~34779 6430 2i65441 6485 . 2!84891'

---~62-SL .... 20.11331: ... ~.---6U1.-. ... 21-10!6.7. -6111 .. 213!-3-''·- .643! -2!.65-44-t- - -- --648!_ .. ~ --2188160· 6282 2086267' 6,32 2~10867 ·~~2 21358!.1: 64~2 ·2~6§613 6412· 2186331'

. 62.lt3 .20.l~!l.f.5.~~--··----~6!33-'------21.1-1f!.47~.-- .. --.A383 i1JJ.Itt --'-4:13 -~- -2i6ft-111- -64~3 2l866tl 6284 2086963: 6!34· 211182?' 63.4 2i3181i 6434 ., •• 36l 6484 218684!: .HSL ... anJ1_t4.3: ________ 6lU ..... .2t1-18!l:.. -63-1-1 2t:Ut0S 643! ··2!:61841 -6485 218804!. 628 20B8QL9 6836 2111899 6Sè6 2137J~t 6436 2i6~1~7 6486 2188809·

. .2lliEid.t..t·_~_Jj!?~~····~iU.US!5 ... -...... -6JI.I7.'-----.2-t-3?-17-t-c------~··-·····'_..J.7-~---Ii~30tS"·-~ -----6-4-8"'~-----2-t--833&•·----6288 2088t63. 6~38 2111955 , •• , 2138$~· 6438 216~569· 6488 2180169·

. ~6.289 .. _. 2.0.8i.Q.'1..J!...~ -- 6s.!9 . --2-ll.t.-3!1' ·-'SU·-···~ -U-39.219 ----~-----6439----~-2~6--.J69 - 64-19--- 2tt-ö t-2t· ·· 6290 2089681. 6840 211!123 63,0 21401~1 6440 2l6f91! 6400 21t6JtJ: 11221 2aio611: 6141 2u.sD'·' ·~ru at.4037t 6441 2t6,-1tt. 6401· at.ts.es'· 6292· 2098815' 6842 2111143 6302 2140'1!J!!I 6442' 2166081 6402 219!.2!1'1' 6293 .20915A7... 6-t43 21-15.14\5 . . 6j~s: 2141213 -·· ·· 6443 2~6f7ÖI 640!< 219Î!J45· 6294 2092363 6.44 211·1·1 63~4 2!.41819 6444 2162061' 6494 21916tt' IJ29l ê_O~~~l~ .. -- __ _ ~84!f 211,323 639J 2142243 .. 6445 2161-731 6405 2l9260'1 6296 zo9sot1· 6146 2t1fB03 6306 214224! 6446 at6884t 6406 2191881·

. ~J9.'l . lQ!:t443. 6Î47. 211.147.5 63·7· 2142843 6447 2l6l84t 6407 2LOS6!11· 6298 20Y3071· 6S48 21!.1675 63~$ 214361l 6448 216.889 6408 2194141:

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Page 69: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

A3 N ... Al

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65~~ 22QS701' 6,66 2129~11 66{~ 2154!79 6666 2!16613- 67t~ 230833l 6!17. .22.0~1e.1~ 6U7 . 2UJ.lt.t.:~.. -'4!' 21S.4i!lt. 6661 2!8êt-t3 '''' 215JU9· ,,ia zzo•6•s· '''8 2aaos19 6•r• aa''''' 6668 2aeta3s 6'1a 23§63st·

-~---~~-----------651-a. - ---:22-0.'-6.85; --------6-t-6.9.---...U-U!-U- ~-------61!-9 2155ttlt- 666-9-- -t,tt-S-1!' - 6119 230 683t 6520 220,021 6170 223188~ &iió: 22!15931 667Ó 228è861' 6?10 2366.3!1'

- -.6521 -~220!1831- ··----'·•-1-1.._ ___ 223224-a--··· ·-- Uit' 22J7&8!--- - --~--71·-· -2!&t867. - 61ft· 23Ö6tt8· · 6522 2206461. 6t72 22~2-13 6ii2 2281467 6612 228~011 6'122 2367611: 6523- 220A7.97 6f73 2212963 6!flJ ··2157467 -6-613 2!8S3'1 6'13 23Q8Sii· 6524 2206797' 6114 2233!187 6614· 22S77Ö7 6674 218~1t!l: 6714 23ö8!177' 6525 -1l0'l!:7!: .. _________ 6-ff5- ----2131587.·-- -- ~JfJ.L---~2-t,i!lJ------- · -66'15- -- 21&.9Jt· · - -6?-t;. ·- 23091f1'' 6526 22084ö5; ,,~, 2233131 66~6 2259435 6676 2281243 6?16 2310665 6522'-----220-~4JJ5.:___ ______ 6_t1l-.... --223AS,63 .... 6f,7----~-2~-IQ.Q'i ···---- 661"1 2216290 -······6ft'1 ---2-31ÖHI 1 ··

6528 2208453- 6118 223!219 ··~·' 2260~~~ 6618 2!8~7,9 61*8 231ió3'1' 6529 2.2.4941.$". 6f.1.9. -213-6083 - ,,._,, 2t61489- 6619 2tlt'1'9 61t! ··!S1j,66~: 6S3Ö 2218-01: 6180 22362"15 6,30 2262035 6610 2!17111 6?30 2312188' 6!5!t~ - 2.!U .. 4-'.!~----- 6111 -213!2J.1 -- ~ ___ :_6,~$;- ~- c2-!62J3S -6611 2!1.011-~ ~ --6-1.--:Ji~-- 15l~1 oi·

.::~~---Iit~;;__ ::_:t i::~::t --------!:1~~--i::i::: :::~ :~:;~:l------- -~:l:~ ---~:i:~:f: 6534 2213153 6f84 22311"11 61$4 22~4123 6684 2!80131 6734 2311231' .,_3JL .a:u~a!n: .&J .. It u3&t!9 . 613-s 2264&43 6615 aaa.oaös 61:Js 2311231 6536 2214Ó21: 6J86 2230563 66~6 2265Ó35 6686 220ti661 6736 231131'1 6.5.37. .2 .. 214.3~9 6187 -2-240.tll------~-----6cU1'- 2265219 6611.. 219666"/ 6':13, 2316160 &538 22148&Jî 6J88 224ot&7 &638 2165127 6688 aatft71:J 6"138 2316841' 653_9 221!55!57 6J8~L .. -2.2-4-0-1.15 . 6t:Jt 226512"1 661-9 -~- --220~617 613! 2317277' 6545 22157~9· At9o 214!627 ''~ó 22&5127 eaoo ealt4tt 674o 2317901 6541 2116421: 6Ji1 2241417 664t 2261267 6601 221@901 ''41· 1319093: 6542 221,,~1- 6j92 2241627 6642 2268Ö83 6602 210-379· 671·· 23!0361' &!J43 221~ 1~1· . --~''-'a --2-2424'-3 . at:.s. . 226B3't &-6o;s.. 2ats'163 61:4! 231-t365"' 6544 2217333 6Î94 2241l15 ,,.. 2268947 6604 2!94419 6744 23t0803: 654!5 _ 221Stot: ---'-1-9.s.. 2241~09 66ÀS- 2269'-23------- 66os eao•4•• 6'145' - --a~s-aot-,,_. &546 za1e!13: 6f96 2244o75 66~6 aa~o'D' 6606 aaojo:~s 6740 232l!Jót· 6154-7 d!21~-Q-3l-- . ___ ,.,,,. 22449!.5 &64'1 . 21'10,01 . 6-601 2!9,4!9· . . . .,., ·-t3ti81!1• 6548 2!19403: ,,,. 2241227 6648 aa+o747 &6oe 2206619 6748 23aieas~ 6540----- .2-2-UU-1:.----·- -- 4t99- -··- 2141221- 6-&~-0----~ --2-21-i-fll--9--- ---· ·6699 2!9;10!- -· ---------6,-i-9·--- -2S!t8tS 655Ö 222876'' 6iOO 2245,07 66JD 22726i9 6760 2!91.363 671Ö 2S2.04S'

Page 70: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

A_l:_ __ J . .J . . .... A.3: -----N-- A3 -~ - - a N- Ai

6?5t 2324ó45' t.Aot 2:!40749 68!it· __ as'~'lU: ~9_Ö1 a4o_i_,_,_.~------- f_9_fi _____ 2142'1139' -~-----~----------- -- --- ---- ------&?si-~ 2~-2.ii~i5:--------~~~~rf2 ____ 23!iói2'9- -------6ii~ ---2316413- 6952 ·a4o i61'1' 69!Ja- 242'19:!1·

~-~ 61 ~! ~ ~21!.?~-~-~-~~-----~J_Q_~~ --~~~t~~'- _ ~~-f~~----.a:~t~1''l ~' ä:s __ .24Di611 ______ --~'-tn __ a•a• 691·- __ ---- -6754 - 2lJ2,461: 6104 2351309 6884 23'1?90!! 6904 240i9t!J 698.. 242981Jt·

-:~;: ~~~~-;Jr- --tl~t i~:it:~ :::: =~~~:~: ::~: -::~:a:; ::::. ~=~~i::· -:~;: --=:t~l~~-~---~----mt----~~=1~:~- --::::-- -~~l~~:l-------- -~:~~ - ~:~=~:~- -:::-:-----=:~-t::;-. -6'159 2~2775.7 6109 23844'1'1 6ÁS9.: __ 2!1'1"'tl!~- ~-6_9_Ö_9 2.4D8tó'1 __ 69Jf:. ____ ,2~st9At __ 1.161; z:J2eoft~ -----,iro ____ 23-sirif:f 6&6o a3aoso o 69!Lo 240~!539 6960, a•:~r~oe~ -~lfli 2~~'·-l~: -- llli_ .23.''--''1 6861.- 23801 .. 2!5. - 6911 240-6519 .6i61- 2L&3t7tJ_,.-

-------------------~-:?t~- ::~l=~_i:__ __;ttl· ___ J_~::~;; _ _ ___ :~::· ___ :~:i:l:~-~~------:_;f~ ___ :::_:::t___ .. ::~t .... J:~:;~:: ---------·-··---___ --··---- _______ ------------~-- :;:;: ___ ii~:;gr ________ J;tt ___ J~;;:~~~-------- __ :t:t ······--~~:i:~=- _____ ::t~ ____ J:tn~~~ . _:::t ~-=:;;:1r __

67&6 t33L98t 6IL6 2357blt· 6866 23833'1! 69i6 a•oijti! 6966 2436t•' .6.7.61.. 2lll98P· 6117 . 2.35l~Jl.5._ .. _ 6!61. 2383883 69{7 240Î7UJ 6967 2436241

6768 2331989 •• ,. 23!11077 6J68 2384045 69i8 241519! 6968 2436171· -~----'-1-~-~------a33~'.!.t _______ ~tf1P .2~-'-160_.9~ _________ 6_lli __ ~_23850I9_ _69i9 . ~24LÖUJJ·__ 6960 2431'13!J'--

6?75 2333477. ••ao 235179? 6~fo 2385'10t 6otó 241§611 '''6 243A!tl 677!. 2333861: . 6_121. -- 236 0189 ... U.?l----~3.8-'.701 6911 . 241.181!-'--- - -- 6.1'11 2430!08• 61'12 233iii5:- . 6122 2360981 61'12 23862!3 6922 241Ï18?' 69'12 243929!1· 6173 2.iJ~~'89~ 6é23 2160181 68;3 238612!!1 6913 241Î!I7t· . 69'13 243063!· 6774 233Sttt· 6124 2361677 68tt 2387693 6914 241i'163 69'14- 244Ö63·· 67.?.5 ... .23.3!9_4_9:___ -------'I2!...23623A9 -- -- '-S1-'-------2U8.J9.3--- .6015• 141--2-'-? - --69-?,'- --2«06~0 6776 233;J73; 6126 2362949 68?~ 2388293 6016 24!~947' 69'1~ 244p'l!l'

. _ _ _ _ __ ___ ~----- __ _ ___ _ ___ 6 7_;•r_ . _aa_3 ___ tJ_f __________ .ual _ .. 23.6.!!11 ,._,.,_: ___ ~a31Ultta_ ___ ' '''- -•~,._,.,_ - _,,_.,_.,.: ___ ~a••1J:Jii 6'178 23375ö9 6128 236359? 6.,8 23$96i3 6918 2~1j09t· 6978 •••ittl: 6779 . a:US445.: ..... ua.t --2-3-64-01 6l?t as•~i3 •••• a•s.ttlt 69'19 2441114?' · 6785 2338445 1 6810 2365205 6$8Ó 23807!7 6930 241'1683: 6915 2441'1'

___ 67_81. - 233~1$9 _________ ,_.,-1~--2-36!105- 6Jj1· .. 23Ct12-'-3-------~6-93-1~----2-4t-197t 691i-··· --24-4~·'-!J" 6782 2348437 6832 2361213 6$82 239!7'13 6032 2~''''' 6982' 2443903'

_ _ ____ ~-~--- ________ 6.1JI3___ 2.IHI4i!I:_ 6&.3_3C-------2-3.660..69_. . .. 6li$. - 2 391989 ---69-3-3-: ..... 241·-9 o'l· 6 983 2 1443901.;.-6'184 as4o,o; 6134 2366645 6•~· 23926!3 6934 ~~1~410 6914 2444613

___ -----·--6'-.B!L- .2!1415.63 1 6tU 2S41~71. 6.85< · 2393419 6935 2411419 691!J 244fH!OI 6786 234276S: 6836 2367581 6é86 2394293 6936 24t~Otl 6916 2446639

·--····---- ....... 6_187'~ ~~--~-!rL!J3_. 6137... .236IIUJ) .. --·-··--'-8Î-7 ...... 2394293 -'-9-37-~--t_.t--ê30' -- 6911- 2446917 6'188 2343389 6138 236192!J •••8 2394!8t 6938 a~èi37t 691ft 2447o•+·

· =~-::- --::::~:~~ :::~ · =~::::-;- ---:::: --=:::::~- ::::--- ::-:r:::" ·:::5 ·:::::::: · &'?ti .2~44.7_:B: &&41 a:ut71t ''~' aso629t 604i . a•••'''. 69tî 244887~· 6'192 2344oat~ 6142 237ot89 6802 2396411 6042 24!~··~- 699a· 24401tl· 6793_ _2_;J~I.l.!IJ1~-------~~~43 2371!77 . - .. --&U!:--239'1-619 6043 2-41~443:; -- ----6tlt3 . 24411'19. 6794 23461,3: &144 23712'1? 6894 2399009 6944 2423731• 6994 2456311• 61_95. __ 2.3~_!'_dJlt'_ __6tj! - 23'72213.. ..6.IOJ·-·-··--2S-9.-90-91J-. -6941 · · 241440-3- ------'90-!J-;- 214!1ló79 6196 2347JD1: ~146 2372815 6806 2399899 6946 2424717 6006 2451!1!19 6'l91 .2\Hlt!l.? .6J.4.1 237288!5 61t'1· 2399627 6941- 2419483: 61J07 ·· 245t87t· 6798 234869]. 6848 23129!J? 6801 2400!0'1 6048 2421713 6098 245i80''

____ 6799_ .2.~-~ftA9_3_; _______ ~6.L4~:..---2.3-14!.9j __ . 680.9-··· I~Ö00-47---------~9-49 241ff6J -60f9-·····24JI801---6155 2349629· 6850 23'1!!1'1 69ÖO 240141'1 6910 24!'1739 7Dbö 2453098

Page 71: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

.. A3. .Jv.

1o ö 1 2453 111· ______ 1 i5_1~--~a~_e_OJJ!J1 ---7ÏB1 a!féLZ.?J'? . _ 7.1-!U. 2B331it3~ 12ll1----- ass&O!l3. --- ----------- ------------ --,ao·2--2-4J46ö7.. .,,,2 248oa39 'toa aso8229 71!12 as:s3eo3. 12ö2 a!S5o"'' ,öo.a 2•s~;'1. ___ ,.ts:s _ a•.a.1:u9 ....... :r.;.u asil.aug .'l.tu ..... a •. ü~-~--- ... ,~.ij~~--a''~s.·~·-·------------··· -··-·············--- -··- ·--··----------,lö 4 --·2'454991: 7154 2481943 1! Ö4 2!JÖ 8219 11!14 2!J!j5!., 7204 2!56 ó ,,., lÖ.Ö,_ .2~5'7'~ 1155: .. 2.48194.3 1ttl 2!48i.Ot '11!Ul 2!Sif9-ês,· 72ó5 2!J6{S61'

-------------~---------- ____________________ ;llt __ t:::_~t:. ______ ~I:_; . ~:_::!!: _____ --~U~- . i:t~:.:t___ ~!:t_ :::t:i::_____ ~~~l-.:::::~:·-1öoa a456at' 1

,,,. 2483.!3 'ia• a!J1o43' '1'8 aB:s?••:s· 1aa8 2!J&t6ot· ------------- -----~----~~~1---}l~~t: f - -- --j1!{-- ~::mt· -- -;tt:---i:ti:~--··-- ~t::-·-i::::r~~------- ~=t~ --C::!::~~

.'1_ 011 24'_ 9.1.67. . .1j61 .. a.._a~s'-__ 1_ • 'iiî 2-Bi2t8t '11tt1 - · aJt,et! ·?tii· 2!J6!Sttt· 7b12 24504ft7 7&62 2486431 '112 2!5~2669 7162 21304!53 '212 296!1321'

__ ___ _________ ..... _____________ 2a1.3 .. 2!4.l1-'~-'-! __ _______ 1 i 6 3.___ 2 48_6 tli.'Z___ ? $Ïll ...... 2!13 lG1 ~ ---- -· 1t6 3- ----- 2~-13-- -···'I~!. · 2-Jt.-,.atJ· · 7014 24614''' ,,,. 2487b37 't14 2!113193 '1164- 2!130033. 7214 286'1111:

---··------------------- --~-'-ii1!. aif.O~-~!___ .. 7l6!.:~ .2487613.:........ 'ti! .... -a!li46!3 ---------'1-'-IJ aUia.•• ------·- 721!;- as6-1s.t1 · 7öi6 2461981: 7f66 24811!7 'ii• 2!1148!13 7166 ~~·~8~1 '12i6 2!J6'12&! 10.1.1. 2~~~J..'t1:.. 1.t.6.1.. 2-.8-~1..3........ .7!-17 .. 2! .. 1!28.5 7167 2-14t-1~1 721'1 2-J6Jl:J!?' 'öie 2462983 '~'' 24884!13- 'iis 2!116t01 7168 2~4l8ö!l· 121e 2!1686!1j·

... !.OiO.. -2•6ttó.t.!L _____ ...... 1t.69.~---2•&Qs,4.9..~------------------'lt9- -···· -2Jt4!D 1 --------------1!69 2!1•i?tJ· - · ···· 1ttt·- ·1!-600!!:.''-· 'Ö25 24643,9· 7ifo 2490301 11j0 2!1~6107 7170 2~4S293 7216 2!169S!&:

---1Ó2-1------2'-''ót1 .. ---- 7f7-2.------2UOJ01-------- --~-i-i1 ---2J~,..·~ ------ ··· ----7-1'11 as•s,.,3 7ati·-- 2!170!:•3· 'öaa 246!159;· 1012 2490397 1122 2!118021 1112 2!14•aot 1212· 2!1'1072!· 'Öa:~.. a•6ts.o•· 'i':J- a•9t.4o5· -'ti a aBi8a•1 ·'l'tJa a"., •• ,. 7tt3 '''ö't~· 1ga• 24&5599 1~14 2492113 '~t4 as19o'' 111• ~~·•oo' '''~ 2111201· -'-02!.----2~6~79i ________ 'lo .. 15 .. -.... Ut.t67'1.. ----------11-fl 212-0-2-29- -- 1!-'!S·-·· ·--···ll••t--&J'-- ··1H!-'--- 2'11 'JU.· --

_____________ ;lit __ J:!~l::, ________ ~c; ...... l:!t:: ____ Jit: ... -~=::!!;_,_____ ;_;_~; ______ ·:::;.;::.. -;::~-----::;::::• ... 7Ö28 !46835J· 7i18 2404141 ?ii8 2521060 1178 2,4,781· 7228 257~685•

. 7029 2'69.55! . '~'9 2414!-41 -1t-t9 2ft1603- 1119 2448!Ö9 12tt· ·2!l,!tUJt 7ij3o ë4&9943. 11eo 2494621 ?t•o 2122~11 118o 2!14.561 723Ö 2115931' 7031 2 .. 4.6C)94!: 1.o.8.1 :UOJ-$8.9 - ?i·31-- 2!123&-31-- '1181-· P40101 -1211· ··2-81,_9~!1· 7ij32 2470t~l: 1@82 a4t6sas· 11~2 2!J23G37 7182 ~·~~''' 7232 211&óst 1o 33 2if1Utt1 .'tg-83. -2A-94tt.s -- -- . 1t-!J3-~- 2!Sa3&a-5-- -- 7183 --~at!ö469 ---------1233 -t!JJI-461: 'ó34 24721!,. 7114 2497l17 'i~4 2,.24573 7114 21Bfi7ót 7234 ast'a''' '!Ö-35 a'il!l.t.: '1-e' 2491981-- ?1_ ''i ·2!Jl!45'1a_ 7115 atti6•t 1235· 2-17'1&o1 7036 2412!11. ' 86 2498701 7tS6 2!124117 7186 215110! 1236 2!7838~ ?.Ö.37 24.l337..5.~ _,,.,_ --2491'101----- ...... 'i~'? -----2!!12543, 711' 2!1Jfi1'ö! -'23~ 2!17--8-9!1'-'Ö38 2474239 7il8 2491811 .,,~. 2~26tó• 7il8 215.989 72~8 2!1,9799'

.7D..3Cl. 2~7~2.:!9__ 7tl9 ......... 2MI9.91'3...... -'1!19 ----2-1!7045 7119 29,S66t 1-239 28,0199· 7o4ö 2474383: ?i9o 2Joost7 1i4o a~274tt 1100 2!JJ.8i:s ?a•ö 2J&oó30· 7Ö4i 2.47~001.. 7i91 2!501509 1i41 2!J28t,40 7101 2 •••• !13 '2•!> 21811'11' 7Ö42 2475583: 7892 250t869 'i42 2!128?49 7102 255,013 7242 288105·· 764:L 2416t!!l~------ 7jt3 -~2!5021!97 'i43. - 252-8.'149 7-i-0.~ 2!'J§'O-l3·· 72_.!-- · 2582381. 1Ó44 2476375! ,,,. 280!813 'i~· 2128989 1104 215~1'81 1244 28819!9 '?Ö~l! ..... 247~JI!!U. _____ 7195 .. -250!8S.3 -'i4!1 ~SJ3-0-!4-1· -7-105 285;-161"---- '24!5-'-- ·-2183&79 7Ö46 2478ó!S3· '~'' 2Jti40!J3 7i46 2131613 7106 2859!61• 7246 258440'' 1Q41- ... 2478655 1

,,,, 280 11!91· 't~' 2131989· '19·? 2,,;,.... 1!47' 2!1844Ó' 7048 2478343 ,... 210!915 7148 2!J32t57 7108 2,,, •• , 1'248 28845!!,·

-- 1g4-9---..t-~t.i.9495 . --'1'-9 -a!l-o-6t6i - . -'.i-~9 213211-3- -71!-? -at~;a8s----- -· 7249' ----asesó1o· 70!50 2!410167' '?îOO 2106261 '1fll0 2!933603 ?200 2!S!J813' '256 28861!9

Page 72: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

_ .Al__ __ ----~---!1.-~-------- ___ A.J_ -~----- -- ---N -- .. AJ- . Jt_ . -- ___ __j-3. - .r.,k - A-J:. -

----~---------------------------------- --;~-;i -~-t: :-~11-r~-------- ~l~-i~---llil~li---------~~li: -- -~~-:~-:~~- -- ---~-!~~ -- -~!:~1~ -------- ----- -~!:t-----:: :::!l------- ----------------------~----------------~--12 5 ~---l~-~~1~-~:_ ___ _!l0~~-----_2_lliiJ_1 ---------~ 3_p _- ____ 2_641Jtc;l2___ _ __ _1~Ö~L- 2 &l.liJi-1. _'ZJI.L ____ J 6 9 liJ,,_..__ __ _

'1254 2~19303; 1804 2615031' '1384 264224'1 '14Ö4 266i43'1 1484 269633'' !!~~ ~_589~JU _ !!0' .. 2.t1.'-.. f.l1 '13i5 26.43.!.11._ 14Ö5 266t1Ö9· '141!!1 2A06SS,. '1256 2989885! 7806 2616423 '1316 2643Sit '1406 267ä2i3 1416 2606433:

- 12s., 4' ~ o 719 · '• o' 26~ "ll23 "'·f· ~ 26t=tlts ,._._;;_'~-- __ a••g-2_ •_ s.: ___ •-----==------------- !4!11-: _____ --_ -_--a ____ -__ 6_ e»'1_·-~-- ••-----------------------------------------------------------~--------------------~---------,258--2,9Ii'i;----==-,;o8----2i0311 -- -t;- s ----- 2644aif-- '1458 as7nar3- "!418 269Aot,.

_________________ ---------~---------- ---------~----.7.2,9 ______ 2_,_~_eb.5_; _________ '!.tao ________ a_u!ln1: ___ -'l!t___ 26'-~2-U.----- ___ ,,J:iJ~-- ~s'1i14.!--- ____ ''''------t-6oa:s,,_. -------------'265 2592015' 7110 2611961 '*'ó 2644263 '14~0 261~411 '146q 26!!9~~-

- _____ 126.1 2.!U2.447_ !111: 26.tlti1 . 'S--4-i 264!l6-C7 74!U. l-'1~0-11 - ·'46~ 2'9R2'3: '1262 2'93311· 1112 2619063 '~'2 2646615 14~2 261~4!3 7462 21QI''~-

___________ 'l263 __ t,9t3..1..L __________ . _____ l~.1.3:. __ 2.61_97___!_5:________ _ 7J.l3, --- 26.68!.! ---------- '-41:!-----c.IU-'5;21--------'14-63: ..... -2'-Ö0'1-S-J. ?264 2593479 78S.4 262Q881 7:,~·· 26413,~5 74~4 26?~6!13 7464· 27til1~, .

.. ----------~------~- _ .. ---~~----------- ~-----'-2..6! .. __ a.!t94:tu,__ _______ ___lus ____ 26212.!-1----- __ --'''-''----··----2648 oo--1-------------'•ts ------2•'•''3: _____ .,._._, ___ .... a7:fu.--8-•-';___ -- . - - ·-'1266 2596]99 7116 2621,,1_ ,~~6 2648!35 14!6 26,f373 1466 21Ö32'3' 1267 2'9.'7-U:.. 'lU'---- 2-'2U21_____ 1~-'1 26.8!535 7 4~ '? l6.7t661 '''' 2751'!3. 7268 2!1961.1. 7~18 2622831 '1368 2648727 74i8 26?6401 7468 27038,~.

___ __ U~!L __ __ . 2!1 ~-~1&1_~----------- J319 ____ a6228!1~-------- 7$89 ___ .26.49199 ------- 14t9 2611.981· 'l-469 - 27ó!JI1'1' 7275 28971,9' 7520 262!119 ,,,ó 26!0839 741ri 26,6981· 7476 2'öJt3''

-~~~ ~=;~;i~~ ~~~i-----i:~!~:t -- ----~l;; -- ~::i~:: ---~::i ~:~:t:i --;:;r i;~::~:~ 127.3 ........ 2!f98ö39 7i23 .. 262J039 7S;3 26S2327 74.Q3 26?f8Ö-I 747:~- 279.419· 7274 25988'9 7824 2625039 13;4 26!1J3431 '1414 2618871 ,.,... 2'1015,13 Ja7 __ !5 ______ 259938~- _____ lii!!L __ ---~6!.1111 ·--~-'-til< _~__,,_a_43.t . 7415. 2686.2!1 '''" 2'P80'1·3·-- -'1216 ~~~9599 1126 262'?211 131& 26!1J3,9t '1416 2885989 7416 27088,3: '21 , ______ tt.Q __ G_s~ ~· ------·-'-'--2' __ __ 2..6.21.!..71__ __ ___ .. 1alt ... 26.14!JJ:.__ ,_,_~__,____ a 6J16.t.t~.-- '·~?--: ___ 2' &a •s.3:. __

------------------·--- -,--218 2601315 1 7S28 2621!1Ji3 ?~;8 26!1J480S 1418 268tth. '?418 2-,091!3. '1219 2~1~3 __ !_,: ' - ... 7J.2i 2621.6.39 7lft 26!556'7. 1429. 2611311: ,.,. 27ó9'7!1J3. ji86 2601315~ '1830 26.29071 '*~0 26!6047 74:SÓ 268~481 1485 27Ö0941 _!281 -~_6l~5_27 ._1 __ !l1 262CJI86J 7 .. 31l1· 26!.6_8U .. __ .. 7431- 2l8~48L .. 74At· . ---2'1111131 7282 2602815 1 1!32 2630111 7382 265163! 7432 2683715 1 7412 27121D!IJ

_______ 12_~3- .. _a~1!.!t __ ~~ . _ __.1!~3_;,t _2l3J1..!_,2_ _,ia~- 26!J?63t _ 1433 __ a••••':! 1483 . t-71t34f· '1284 2603847 1~34 2631951 ~~84 2651961 7434 268j603 7414- 271134!1J•

___ ?.atl! 2'---0.~ .. 42.3 __ ____ .7.i:S5.. U3tf.!1.--- 7$Á!L 26588-:St -- 14-3!5 26&jO:S3. 141!5 21!.@11!. · '1286 2~015~3 '1!36 2632383 7386 26S94!1JS 7436 268f2•3 7416 27tS731

_____ ?28"! 26_D __ , __ 5.fi3_, . ..!W_ .26331'--1-:. ........... 7ll-"----2AS960B------- ... 14-37- --2617~61: ..... 1417· ·2'1-!.-113'' 1se8 2605&95' 7838 2633535 1sàe 26s9eos '1438 268'3'' "!488 21!38'' ll.Bi 261111-!1,_'. '-lllf _____ 263396_'1----·-- ~, •• ----2-6-60'1'1!5 - 74~9~ 268;31'7 '1-489 2114111' 729ij 26084ö7 1140 2634207 73~0 2661399 7440 268fS89 '140ij i!lStli· ?291 26086,5;. 7i4S. 26!4915 '13P1 2661309 7441 268f069 7401 2~16061 '1292 2608605' '1!42· 2631509 '1302 2661495 ?'442 261A81!' 7402 2116209

. 72 __ 93... -- 260 9_2:lj,_;____ .'1i4J __ .2.6!1599 ,:1.3- --2661879 '-4•3 26lt319 7403 27! '11!: ;294 a&o•e•?: '1S44 2631,,1 ~304 26632'1 '1444 2ee~"!&l 7404 211~;o3:

. _729..!. ____ 2,tJO 9 $4_1 _ _______ . _'184!5· _____ 2l3U'l5 _ ----'315 .. 26.63695 -1--448 26.o!IJS: '740'-'--- --2'71 7'ts: -'1296 2609943: '1146 2631111 73~6 268312! ?'446 269i8ó1- 7406 2'118461 1

1.227 .......... ll104~L.. .1-1~1 2.638095 t3,a'1 2644'110 '1441 260t801 ?40'1 2'104,1, ?298 2611143. 7S48 2631383 7308 2665'149 7448 269t3d!IJ 7408 2710953-

·- _ ____________ _?299 _______ ___2.611983: ________ 7j49 __ f_6_3Q.7 .. D-3 --------"3U= -- -26-U-'1-49- -- 74-4-9-- -- 26030--'73---- ·· 1409----- 2'2Ö1411

7:5oö 2612319 7S5o 2640543 74öo 2666349 ?4JO 26t35'17 75Öo 2'721041·

Page 73: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

~~~~~~-~~-~~~ ~''~L __ 2Z~~~i'flt: 11,1_ 21~8!593 _ --+•21 27!66Jt? . . 1.6!!11 210.31'1 ''~di!~ . aa:st~1!h ·-···~~~---···---~-········~---~----···--· ·-·-····-··-- ~---;;~~i-~----~;~~-: i .• ~... _;1;~_ -~1~_:;::: ........................ 11&~- .. ~;l~g:i•. . .... I~:i.. .:~~i::t~. ;;t=·~--··-~=:t::r~----·

'1!5Ö4 2723óft; 1,54 2?5Ó393 ~6~4 21771!55 7614 2IO~Ob1' 11Ö4 213{84!' 1'0' a1a:s~•9 111' 21!n6'.1 ''~' 211etss ''!L5 aaoJOiJ'1· ''ós' aa:sa111i 15~6 212:~~1 '''' 2'51545 7606 at7919S 16!6 280§!83 ''~t 2134ó~f·

---~·-··-~·-·-----· ___ ~----------- _______ _::_~ __ 75D.7 ___ __1Z_~-"'O_b1: _____ 1111 ___ 22~2.11.1 -----..fi!7 2119-&tt--- - '16!1-1-- --28.0106-J.:... .. _ ... '1101' -tl344tt· 1558 2?254ö9· ,.,. 2'152529 '1608 2179129 '16!8 2108110 '1'1ó8 ~è344t1

___________ ,_5.0.9 .. ~-- __ a?.at.aas• ______ ___.,JJi__ .225JU2_ ·····--~-+i.L-----2'-aoJtt~------- _,_,._g_ ____ aaajas• ------- "'"'Q! - ti3J6tt· '1510 2726891' ,,,0 2152913 '610 2'18126~ '1660 2808983- 1110 2836'163 ?sis. a12A$~,- '''1 21!33~.3. ''it- 2711243. 7661 aatna,o· ''iî. a•3•,••: 1512 2?27185' '1162 2754!.13 7612 2781743 '1662 aa1ö'1tt· 1112 2837oos:

-~---"~U. __ :t ~7-~'J_tl_l_: _____ . __ _1JJJ, _____ 2'Z.5_4t_l9____ '?6{! ____ 2.1.82.:519 . --- .. 7.663 --2-11Ö1.t1· ~-1113· ·- •liU1JII '1514 27288•1: 7964 2'154929 7114 2~831!1 1664 21tft1i1 71,4 2AS86SI

·------------------·----- ____ __j__S_i, . _ 2:11ct Sll_:___ ~ _ .1.1l1. .. __ 215646'---- .. .,6.1!~ __ !1831!1~ -~-------'1..66!5 US.-ttl1 1-~1J 2430-06'' ~5it t'1zt~&1: '~'6 2757329 ?6i6 27A315t 7666 28tioss· 7ti6 28304b:J:

.. (~j' .. 2~~Q ~e~ .. .. Iltl 2?5ll.l9_ _ 761'1. ..211438! .7.667 iJI;tJII' 1111· 2130911' 15t8 2731!61: ''68 2751~'13 ?6i8 2784111 7668 281-910 '11,'8 2A4g11~·

...... ~-·--~----~ .. -- _j_!1_l__ __ 2'1J_1.3.l_l;_ ________ 'fl6~---·!_'l!_li!~J, _________ .1,19 ..... ~'!&6815 '!66Cl 281!1-71!1 · · ~-"-1-lÇ\L 21407,1· 7525 2'1316~1: 7j7Q 2'159063 76ió 2j863Öi! 76~0 281~0ti· 11,5 214tDi1: 752i---.. 2,.3246J.; ___ :__ __ .~ .. 7171.-2.151l9!L. ______ ,,j.1---2'110IS - ---'611 -21s,40i!- · '11ti"- 214139!'-· 7522 2732753 ,,,2 2750735 76i2 2'188205 7672 211f103 .,,,2 2843123-7523. a'l3:SJ9l; '117'3 e'•os.o~ '.613- 2718295 16-'13 21!.1'·'-!· 7713 2143:sst· 7524 2'134313 ,,,. 2'161631 ''*' 2'188535 1614 2816101 7114 2843843

... 1!12.!-- .2.134_19!,_ --~-.te~''- ---2.'-14431 -------'!-~a.: - a.1a•••., 1''' aas.!ots.:- .,,,,,__ 2144!161 ·-7526 2736Q89 ,,,6 2'16!901 ?.,, 2'1903,9 16'16 281!005· ,,., 28413''

_________ 7!52'! .. _..2!3to.ei ---~----'119'1 ___ _u_uttol ~-~·~~-- .,,,, ..... 2'on.rto.a. - ,_,_,_.,__. a at ''Ot---- .,, • ., .. - 2141119· . 7528 27363Ö5 1 7f?8 2?63143 ,... 21909!~ 1618 281.117 '1118 2~414'11 1!2.9. - .!13Z025: .1f?9 27.6t4.!7 1 .. ~ 2712863 76'10 2Alftl1 .. '''! ·2-846GOIJ: ?536 2731889 7110 2764!.17 ?6~0 2792543 1680 281t1!5 '1?30 284t811 1

. --~--- 1531 _ 2.2'384_65~ _____ ____l_l_ll_ 2..'1UU9 1-631-' 2'19254-3~-~. 76ltc .. 2-8!.8703;. ·· ----- '1731· - --2147111 7532 2738Bth~ 1112· 2'16,1413 76~2 2'192759 7682 ~!81.86'1 1'132 284787, 1

______ _!5_33 __ ~-..21_3.?_4.2'~ . -·~!ff_3, __ 2.76!1t-lJ ____ .... ''-',s; __ 2'193135~ .. 1613- ---24tti!JIJ - ---'113:! . -214113-Sj 7!534 2739857 1t84 2761663 '16S4· 219477!5 '1684 281Ö331 7'134· 2849931 _!"~~ 2~~--p-~~7: ~l.l_·~ 2?6604.'1._ 1,!,38 . 279525!! '161!1J 21.2{48? 7735 2840930 7536 273~8!51 ?i86 276'1717 ,,,, 2195591 7686 28130'11' 7736 2850563'

... 7.!!3'1. 2'74P~! __ l~_: .!.ltt~ .. 2168i!.O:L_~ tl6rJ'1~----21961!.t 7681 . 281!071: .. _.,_,,,_ 18'-ÖtO,,_ 7538 2741105 7988 2768405 76~8 2'196615 1618 281,461' 1138 21,1411' 1539 .... 2'1 41_l'l1~- . l.ll.9~-·- .276''''-~----- . -'63.0 -- _279_6695 7689 282442'1 - ,_,.,, 2851311'· 7s4ö- 2741969 '''o 2770535 16~ö 2797367 76oo aaeiat1 7''Ö aasa,at· '1541 27431!t5t 7,91 2770535 7641 '279811! 1601 212!1843 .,,.1 1853083 ,.~i j743J.j! ,.,2 27'10871 ,6~1 2199ti9 7602 2116083 '''2 28539~~ 7543 . 274398$.~~~- ''93. 2'111-C"!5 __ 7643 2'99115 7603. 2826.!8.7: 174-3~ 185394,. '1544 --- 2'144753 ,,,, 2172191 1644 27998!5 7604 281;400 ,,.. 2854015' 1545__ 2'14!16!7'____ _.,,,,. .!7.1!~55 . ~ ____ ,_,.I!J 28Ö-Ot09 ,,,_, 282·····- - ,,._5-c .. -285!1271: ~-,546 2746385 1

,,,, 277!415 '646 28Ö1i51 1606 282f61t 1146 285,945· 7547. .21-46,_4.5-; -1J97 27?!,CI9 .,6.? 28Q1t59 7607 ·l!ttt6Ö3 1141 28561tt" 7548 2'146745 1 7998 2'1'1480? 7648 2161255 1608 282~Ó83 7748 2156908· 75-4!----2!!41585'-· -----~-~7J99----27-1-4-807- - -?6-f-9 28Ó2641~ - 76!~--- 28:2t7~t·---- ... 71•!' -~~-tlfl80~' ''50 2748593: '1800 2?'14927 '1~t0 280384'1 1700 2835403 '1'50 28,86~3:

··~· --·-·~-.. -·-· . .... . ··~-·· ... --~- --~~ ...

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--------------------~--~--- ________________________ H '---~-:............ N A_3_ _ ____ N__ .. A .. ! .t:L. . . ---~! .N -41,

1751 28S8633: 7101 2886317 11!11. 2.~~~-~-1- ______ _1_9Ö.J.~- _2t_fiH1.___ _ '?_9lL __ 2!70283: ___ _ ----------------------------------- --- ---- ~-------~-,,52-~-2~''o-rr---------,ia2·--2se724i ------,ü2 a91,4sas 79öa 294~049 ?982 201ös's.. --- -·----------------------·-··· --. --······--------------!., 5 3_____ 2.~~~3_f_,_-_ _______ .,jQ__3 __ U8.1.8. . .4.9_ - -- _.,_._~tlè -. .aO..i.~I~'L . . ........• ,_ t1i3. -- . 2. UO~'------ . ··- .7.913~----29" S:ó ,,. -----

1154 28.60061: 7104 2881849 78!4 2016111 7954 294S31S 7984 217162' 1.2.55. ... 2~64.315: . lJ.O! .... 28A.t.00.1 - 'lt!~ 29~61t7 19.§5.: 2t-4jl61,.-: -7911- 2SI;I613' 1156 28.615~7' 7106 2880625 ,.,, 2917!33 7906 2044811 1956 297!211

_____________________ - ______ n~z ____ ae.§_2_1e7 _______ !!_tl. __ .2tu6H __ ~ti.$-z ___ 29-1!8~•- ____ ,;.§~'1~-2•••:smt _____ .. 1.9ti .... .a.t7Ja:s,J __________ _ 1158 286265. ?' 7108 2880'145 78!18 2018,85 7908 204876S: 7088 -2~07'4708'

__________________________________________ !Z!i9 _____ 2_!!~_6_t7: ... ____ _z_u_t ______ ae~.t.4J.1 ___ . . --'-U'-----~2.t~9ou~--------1.t_ij.9.~--2t4t4.!1.: _______ ,tll!------2t-'1~J.Ot· -------7765 2862857' 764.0 289t089 'IAO 20!948! 7910 20C718S: 7060 297!013:

____ ,_,_61 2~i.4ói~- .. ___ 1.1.1.1: _ 211.1n'1 "-üt .... atant.O!L 7911 -~ut.h . .ss. J't6i· 2t'''''' 7762 2$646S3 7112 2892177 7161 2921041 7012 204;201 7961' 2976123:

________ 11~:! ..... 2!6t'_1!1.: ______ 11.1l ____ 2,12865· ______ 7IAJ_c. __ 2.9.11.D.U~--------''-i3 ... :.2ftl161:-----..:1.1.63: ... :al.t!UI• __ -------------7764 2861449' 7114 2193849 78~4 2011473 79i4 214~01!· ,.,. 2976831

--------------------------·- _________________________ __!Z_6_1_ l!6 6j!~ _______ 1_1ll_ __ H 1_3_14 Cl ________ 1AA!~ __ c ..!U.2.12t. ___________ 7.91! . ----UA.t!6 1- -- ---l-16-S '- -2t1UA1·- ... 1'166 !S676ö9· 7116 2194!85 1866 2923'119 i'Ot6 294t9i3 ,966· 2978785· !.'l6i a~6.:7.6.d.~~ 111.1 a.at!.l.8.1 ... _._ '-*-'' 2t23it:s ''t' z••ö6At· . '-'6' 2,,,,,,. 7768 286'1~73: 7118 2896105 ,,68 2023903 79t8 215~1!.3 1968 297881~:

______________ ... ____ '1.1.69___ 286!'_1lt''----- ____ 1_1l9~ ___ 28964.2 _______ .1.Ji.t ___ ....2J.J48.!7. _________ 7Ji9 .... atsiû3 . ..lt6.t· ..... 217t•s.-t·-77?Ö 2869703- 7110 2896777 71l0 2925t93 7910 201{1!3 ?97ij 2t8Ó51~· 7Z..?t.:___ _l~l Q_a !_I;__ . _ . _'/811 ____ aL97_o 29 _ __ ~--·--J 111 :....2.ta5..tt :s _. -·- _, Jt.1. 2 o Jt-'''~ _ ..... 1t!lt-- 2 • a1 ruu.·- . .,.,,a 2870225' 1122 2199417 'ht2 202524! '''2 a•sie•s ?t!i· 2t8tJst 1.'113... 211711~5L_ 1113 2.891.11 '1..81!; .. 212.5769 791:! 21!S2,t· 7013- 20811tfl0 7774 2811~Ö5' 7124 2898607 11;4 201?109 7924 2tSS!{I 7174 21835'19

.11(_! _ _J!_l19.tii.L ___ _1_U.I ... .2Atl!.69 ........ --'1.à.11-'--·· 2.1.280-1!. 79.1!5-- -·20-541-!9-:---- ---19'1~L- --218387.0 .,,,6 !872217: 7116 2000319 ,.,, 2028113 ?026 20ii4i~· ,,,, 208412, _________ .,,.,.l ____ a!7a&.o_,. -------------'-117 .. -2-l.ó-M&s. ___ , __ ._;_., a•t•-O-t-'1~- . --'9-1'1--- aosi4-t.-J'- '''': -tt8468t 1118 2873311: 1aaa 2toa9t3 18'à' 2129729 7928 2t5i7b3 1978 ateit63: ,.,,._ . 28?.38,91. . 1tao ató2,05 ?Jil aoa9'12t 792J: ·a•th.41:- 197! -21861!' (18~ 287408~ 7130 2903473 !$,Q 1 2030353 7930 205?815 791~ 218.41t

__ 1781 - 2.1H,.193: ........ !l-3.1 29Q.SJ"73 1.1-1--- 2131,0.8'-- -- . 70S!--- -21,J87t· ---7011· . -'-2f87J8;s: 7782 2875673: 7i32 210:5857 1882' 2032033 7932 2058871 7082· 208800#,

____ ,,_Et:~ _2_~1_,_&1.3, . _.,_._33~-- .a9n,A.u ___ .. . 7aa:s.~ ___ 203244t ''s3 - -. .aoto3R1 --- _,.,13 2o.saóot, 1784 2876489 7i34 20ó40I5 7èi4 2932115 7934 2965863 7014 aoaat&3

. 1785 tll'l.iAf· .1U.S .... UO!~OS: - ~--~ 20:S33p!J 10!!1 206~863 1989 208!4~1> 1186 2178ól§• 1136 2001705 ?.,6 2034409 70!6 296~863 798~ 200QI6S

....... J18 .. 1 21786.9_7_:__ _ __ 78JS7 ..... 290!913 .. --------"!$17--2-1-34.4-0-0 1-931 --206!961' --19-81' 2t.O-G-11t' 1788 2178889 7138 200,241 1888 2034553 7938 296t9SO 7918 290!281:

_ 'llS~l 21793t.!~-- 7&3.9 .... 2to7241------- '••• ----2-036oao - 7919 aoe34t.t ?989 2t9ia•o· 7795 2180833. '1~40 20Ö7~09 7190 2936665 1940 296~899 7016 2092$39 779i. 2fHU Bl!l:__ '''"' atoaa41 11~1 ati6tJIJ 794! 2t6;oo:s ?tt! 2t9t:no, 7792 2888!~3 7142 2909137 ?801 293?f~7 ?042 2 •• , ... 9 7902· 2t92fll

___________ _ 1J'93_ _ aeaa.4ö~·---~ .71.4:S .. __ alo 9!521 __ ---'813- at38.S!J7_ -- --'-94~ 2-0U-4-Jt '''3- 2103441:. '''4 zaaao93 '*44 2ooo9os +8o4 2o~ae•9 7944 2t6•o:ss 1o•• 299!JD'!

.. 179!... 21l~l17.________ '114! .... ..2.lttJ673 --tlt!J ·· 2JP--a-4t- --- -- '--9-41·-- 206.J.!.I-:--- -- - - 1-9-0S'-- 20-9S4!Jt . .,,,, 2884211' 7146 2911369 , •• , 2939233 1946 21$;6~9 1906 2095.59

.. 112.; ____ at.t.~6.97 _,,_._,. 2ts.tS69 - ta.o1· 2139111 . 7947 2'''''9 '''' 2t961'1S· 1198 Z885117 1148 2911,29 1808 2040289 1948 206.147 7998 2996111·

'19.9 .... 21.8~1,!7 .. ______ ,.49.· .. -2.9-t.-114-01 - .. ., .... o. 20408-1:_1----- .. --19-49----19602'·!'·-- ·19t9"- --i99681i'-·· ?&55 zaS,J!3 7&5o 2013961 7oió 2040St7 10!o 2016283 aoó~ 299609!

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-·---------~--------- --------------------~------- --- ---------~-------- ---·-- ------------------

Page 77: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

. A3

-~--~~-·-- -~-~-~--~~- ---~------ --~---~~*i------~}:~!li~:-- --- --:::i-· ~~t:!~~ - ---:1fi:----~~:i-~i~· :::i· ~:~1~:-;------:~-~~- -- ~:~ ~::;. -------- ---~--------------- ~~~~---- --t:i!~~--~:l~t~-r.---~ ~-lil~-- · ~:ti:ti ---- -:~:~--- ~~-:i:!: :::!: ·-~r,t:!;---- ~----:t~r----~!~t:::, --~-. ~-'ö' _ :saa6,S.t .. . ~f~!!l :n1•o•1 ~.a .. s 3.:!l3J.4.t &6.ts 337iBI7' a., os :s.aotoe•

esi6 3217237 Bf56 33162D3 e&§~ 3344133 &6!6 337t''' B'lti~ :s•U~at~ -~--------~----~------~-------~-- _____________ .J~,_a, ______ la~t~~~'------------'-j_!f_'l __ ~ __ 3.3t6'l!:s__ _ ___ &.Ao!~----3U4.t!.J.. _________ ,6_., ___ JJ,40.Jt---------•'o-'~--u-öU'1t---·----

e,ge 32876&9 etse 331131? ••a• 3345!~3 &6!8 33'ltOS9 8'Q8 34Q4óst· 09 . _3~8!2_~3:~ ----~l,_i ___ ll113~l___ __ .860 9 · _3U6.t'!J. ________ A6.Ji ....... .c33'lt!JI'f · .&7co~ --~--;.l4-0-U-1Iè __ _

B5iö !280445' 8,60 3317357 B&io, 3347933 8660 337.923 B?i.Ö 34Ö!I09' _ __ _ ___ ~511 3~1~4~'-~ . '161.. .33.1.1~13 ... . .. IÀ!1 334826.9. 8661- 3S,67:St 8''t· S4öl101

85±2 !28Y403: 8,62 3318803 86i2 33486!3 8662 33'ff267 8,12 34ÖI691

----------------!~1!------J~-:t~lJ·.-- --~---·-:: 2:: ~- ~ti~l~} ~----- -:tt ~. ·~~:::~i ---· ~-~-:::!-----~:;mr · : ~t: ·-·-- ~:1:::~: ___ _ ___ ----·-- --~~-~~is _____ ;ti~lf~l .. ___ l~6, ______ 3l!..l6_IL ________ S!i~~ 3.3.9949 . - -·--SA6B-·---U'-U'-1~ 871!5 -3.AÖIJ03-

85i6 3292229' 8'66 3321509 8616 33S0429 8666 337fS41 8'116 34Ö867~· ...... EHti? . ~a~l~Ö.1: ll.t7 3..:U .. 15Jt'l........ S!Ï1. 33!1ll09. 8661 .. 338ÖS.t5• 8'i7 34Ö0481'

85~8 3293405' 8~68 33~1989 !6Ï8 33S1J2S 8668 3380195 8'18 3410011' - A5.19 ___ !29~_jjL____ _ __ 8169._ _ __ 3.322!6.5. -~----8-&Ï-1- -- 33523.9 - 8669 . - -3~ti6tt-~ . 1119 ..... . 3410 015·

8526 3293789' 8t70 332,381 86ió 3352637 86?0 3~8j411 8110 3410399 8!2.!. ..... 3.21~_4.6.1.:. . _.8!71.: 3!2j2.4S ·--·-~-!.tH--3.35.31}.81 8671 3311417:- -----8721 -:U1!1S~· 852* 32955~1: 8172 332.245 .,.2 3354941 86~2 338.,1, 8122 341222~ 8!23 . :S295J.QJ..:. . 8!73 33.2!133 86-f:S: -l3J4.t4t I6'J3 338~1·7· 8713 3411895· 8524 329621!: 8'74 332,757 86~4 3355t57 8674 3383867 8;14· 3413311

_8!52.! .. __ Ui1.!1l1; ____ ··- _8J1.J.~ __ 3.32 5~!51.. -- --···---86èJ:.. __ ... ~U-J.5.1S-3--·--·---·····-- .&611 · ·· 33&~'197: ~-··~·-·~-1'115- ~-Ut-."6''· 8526 3!9&9öt: at'' 3321925 &6~6 3356309 8676 338,417' a,.,. ~41toss

.8.5.2"/· ........ 3.2.9.8i0.1.:~-~-~~U1-'-------3.32l0'17. e6i'1 --3-».6-41-1--- .. ---~16'1'1· -· ~.JJ-.8-?'&7· ·--· · - -8-?t?·--&•it-&-95• 8528 sa99o4J: Bt?a 3321461 &6i8 33!7tos 8618 33B;oit '''$ 341!3tti 8!29 329~Ul1'. 51~9. . 33~1.469 -16-.t· ~~·-~«J. .. -· ~ 16_,9- · -3"!0!.1 · 8719 · --J•it8~t~ 853' 3309389 e,eo 332tB53 a•~o· 33S~t4t 868o 331?2b3 8736 341''''

. 8531 33Q!I.!UU' .8J.8.!1. ... - ... 3329!J .. 9-'! .. - 86-tU 33!J9-!~-t--~---8681C-·~---3-381~'1' 8?:Ji· -i!•H.:ff-&-:J:---8532 !3011~5! 8912 33~ó365 ej~2- 3319381. 8682 338.883 8732· 341,983.

------------ _8533.. 3302l41:. 8_~83.--~33036.5 ..... 8633 3360t40-- .. -----8613---331f.261 8113 34!,-861-$-8534 33036~5: 8f84 3330701 8fS4 3361205 8684 338!603 8'34 34111&1,

. 8JS!!L 330.3641: . ss.as.; . 3.3-3-tt.sa. ••~• -336tsao '''' 339\:!1.8'1 8''' 341156,. 8536 3303989 8J86 333307? 861. 3361589 8686 3tftt0! 8,3~ 3419~!51' .SJl31 . _ .. l.:U47J19 8117 . ~33-335o.9-----·-••1'1 33623!7 86t'1- . -33tilt!l 8'131 ·· 341ói-A1' B53B 330B6tt: e,as 3333845 86~8 3362,41 8688 33tt1'' 8138 3421011 1

8539-____ l30_tQ.29:... 8fl9 . .333A0Jl1 _ ... - - 8610- 3362741: . 8619 .. 3191'113- . 8110 3422131 8546 33o&ót~ e•oo 3331525· 864ö 33~2,•t 86to 33t3683 8;•o s•214t3

.... 8541 3~Ut1~.~ 8191. 3331!525: &64!1 33636!3 8601 330436-1 114i 3423931 1

8542 330722~ a,o2 3331&69 8642 3364661 8692 3304&41 8142' 3424709 8543 _ 13o.Zaa.t·.. 8193. 33!6!149 . ..86~-3---3.3.65045 8613 .:ssti1-'l'-------8-74a-; .... 34!410t~ . 8544 !30746Y' 8194 33~7781 86~4 3365477 8614 31t6t13 8744 a•a•ao3: 8.54!_ .. .330~1.W.i ___ --~- -- 819! --. 3331261~- -..86-45~~ 31661-tt 861!1 331601! ----- 8J.4-! --34t-J8'19' 8546 33096'17' 8f06 3331597 8646 3367781 8606 330+601 8746 3426117' 8!547 .3310.0.6.1;. --- lt17. 3331341 1647 3367'18t 8697 :!SfÀ!-6-! 8t4'' 342JDS!: 8548 3319253- 8f98 3330965 86~8 33684~3 8698 339~113 8148 341,35t:

_ ···---·--···---·------------- ·----·---· ·-·----- ----·-···- ___ 8549 .. ___ :s:u1_s~7·.~-------~-'-''--. __ 3:J!t965. 8640- 33-''••~----·-- .. 16oo 3390!163·· . ''4-o - -34!'1917:--. -855Ö 3313169 8iOO 3340565 86SO 3370043 8700 3390563 8785 3420117'

Page 78: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

··~~--------~----· ______ ____ ___________ _ _________ ft. __ . . . .!.3 -·- .. N. .U.. ----~ . A3 . -N - .U.: - - - N A3:

--··-~------------------ __ .Jt1~L __ 3_~t~U7_"_ _______ li.D.1 _ -~A!U.263__ _________ ~------3J8.82f! .. 89Ö1 --l11i.1-A~-----AU..t- -314'189,. 81~2 !~2919~' 8102 34~941~ •• ,2 34887i1 89§2 311.213 80!2 3847933

-----------~----·~---------·-··----·---------- 8 1~1 ____ _l~-~JUI!1~---------lJ03_: ___ 34.50631___ . _I.J:S __ JA8,9.3l3......... _ .19 .. Q3 ......... .3!11t2.S!!----------ll!3~-----U414&-1~ ··-8754 :s•3t415' 8$04 3460309 8814 3490333 8904 381Î773 80!14- 3140,3S .875S 3~U.i35l ... 8 .. ~.0-5 3-A61$59 aejs -3-,90-l-!S- 89Q.5 3!tÎP-0 .. ..8JJJ 3-JBi-J&I' 8756 3~J2685' 8 06 3462031 88!16 3400009 8006 311n911 80!6 3!15bSOI

------·-··-------------------------·---------------a;;.:---- -·~:~;~~i+-----~IJ!i-----t~t~&Ji ··-·- ::tr- -~::i~=;~-------:~-äi-- :::t:-g;~-------- ---::::----1::1~::· -··-------- ·-----·-·-···------------------------------·---···-----~8··.;;;; 50 34 ~3~~ ~: ___________ !t0.~---~~6-'Jll~-- _ . _ !'J;9 l~t2_~_8_t _____________ ,_9Ö9 3U~3. 8.11!~- .3151!-~!1-- -

~6 3434t43: e&to 346428? 8eao: 3402,8t atto 3!114861 896o 3151013: . _ ~-~6_1 . ~~1~717 lt11~ --~~46.f_a~z e1•1· 3~,~t~l7 . ••11 . _. 3.!~&69 .... et•~: 315!1'''

8762 3436111: 8112 346,~61 8861 34t5t09 89!2 3!12108$ 8062· 3J5SI49· ·--·-·---·--------~~---------~7~:, .. ..3.~.-~.~~~7·~---------~-ts.~-----~~-•6J_1!~~--------~----ll.3 ____ 349-'!.09 ________ l9.i1L ___ 3!t.!U:,_ 8 96!.... 31541.,!:. ___ _

8764 3436617 8$14 34668!5' •• ,. 3495373 19i4 311~281 8964- 3~54821 ------·-·---------·-·--- _ --------··--------·-~----·------·--·~~..L~5 _____ ~-~~I~~i __ ~--JlU..-_ __ ___ 3 4 6..tll~~-----------~-i.6.!.. 3.416 !1.3 _________ e_f1 !5.--~ .. cllltBiS -- ... 89!5 "'- -3.15llut~----·------- -···---·--· - ····----

8766 34!1805• 8816 3466855· 8866 3497481 89t6 3111380 8966 3156813' ... B.'t67. ___ !~~~-''-~ . '&ll_ ~-3461.112.. .88-J7 ·3498tS3 8tf7 3.!11;917' - 8961· 35568&3:

8768 3438991: 8818 346&os5· s••8 3498469 8ti8 381t06t· 8968 3851oos, -···-··---------'-1.~--2-----~-l•~ei~_z __________ .. li1Cl _____ _3U!ll1~----- .U.f-t ______ :s_,~-'!.0'.:. _ ---~'i' :su~~-·----·-·•••!· :ssseos.s: -

87?0 34485,5 8820 346073~ 8870 3!00317 8020 3!110309 8970 3158913' I 1.1 i .. __ . ___ 34 !li~5_L _________ ,_!_2.1 ___ 34'lÓ.287___ _ __ A.fu: __ _3_!io3l'l-~ ... 8921. __ 3!1[!7.1'-----~·-····.8.1'1-i.·-- -31!10411 i~i2 344L94!. 8822 3471015 88f2 3!10629 8912 3S.i71' 891~ 3159711· 87(3 3~.414.1.3 .. _ ~ 8l.i3 347~055 8173. 3!g13j9 8.913 3~1112!.3-- 8173: lê6Q41t .. 8774 344~439' 8124 3411439 11;4. 3Sg20!3 8914 3!13,213: 8914 31!6~31?

__ ....... ____ 871!~ ____ 3_444U.:1_'_ ________ 8.t!S .. :. ___ .J,7.t!l.83 .......... Ut-.L------U-0-lS-0-9 ........... ~89.2.5- -3lJtAlll- -_____ U,,_., ----31-6t$t-f' ····· ---------·--·-----------------·- --- - 87?6 3~44813: 8116 347!791 BI!& 3Sij488!S 89116 3PJ3lt461 89'1~ 3S6i,IJt· ___ ---~--- ----------------·· --····----~·--···--8.7.!7 _______ 3JL4J.8.3..1; _____ uu ___ J.4~.401-'--~--------.8U7·------3144J.U ... ~----------A911- -3~11346t-·-----------4.0'?--7.:----3J-6I06t'~

87?8 34464ö7' 8828 3474Q79 88,8 3Sij4940. 8918 383~149 8018 3161873 8.7!9.... 3~4616! ........ Atai.. J4.t!SV3 . Al-tf ·lfi4,49 8-019-- -3JÛ46-9' -8-9'!"-- -3!J6S1t.J' 8785 34471,1: 8810 3476119 8810· 3SQ504!S 8930 3!135!25 8980 3863833

___________________ .87J~i __ 3.44§.111: -------~--~.8$.3.1-'---34'1.lt-1.1i --····· 8$.1 · ... 3S.Q'S'-U-----A.9-3-1..: _____ ":JQUI3~-·-·- --- --8.0-1-î~ --3S-61-P3;---8782 3448447' 8832 3476215 8 •• 2 3SQ6677 8032 3'3f41S· 8982 35662ot· 8783 .. 3448447'___ _!IJ5_3 ___ ___3_f.!il._3~ 8A83 3!0744!:. _______ 8Jl!3 .. ~--3S3,31S· 89.!3- 3S-6620t'--- ---8'184-- 344&sd7 8834 3477895 8114 3SÖ166t 89!4 3B3828!J 8914 3!!161145

______ JtlJt!!L __ 3~~~-4.:79 .. -·- -- 1l3! -- .. 3-U.S-2.7...9 .. - .... ..... 8tt5 l!IÖ84 "/7 .!031- 313t21!1 8915 316712{ 878t 3•50717 !136 3471S73 •••• 3!ij9t07 8936 3•3i603 8086 3868701 878'1. .... 34,10.S..J:___ .8ê:S.'l.~-.!UI!O.t ________ ___._U'1-~ ·----3-SOl,t'l- --1931--~3-S-It26-1---- ---89-8"1· 31,04113:--

-- -------------~--------é1ee-- 345&351: ea:sa 34ea1oo 8a8e :sso98it a93s 3ssïaos 8oae 3!16t41s. ______ B7Jl9 ____ 3.\tf.iA.U.: ____ . . 8!10 - 3.48dt0.1-- -------tUt--- -3!111-111- ·· -·- 8930---314§.-IÖ! -------- ····· 89-19 311j\9S:

8796 34,31,5 8&40 3480711 8800 3!11159"/ 8940 3!14t0t3. 899Ö 31,iól1' .. 819t 31131.,.5: et.41 34-!.t.-1.3.3- -UJ-1· 3!Jt21f!J 1941 JhiUO lOt~> 39,iÖ!I7''

8792 3453607' 8642 3482245 8802 311236~ 8042 3!141041 1902 3!"1t1!1J. ~- -----·-- ________ .... ..879!_ ~~-J--~.32.1 ........... ~.8643 . 34828.69 -···-~-----Uri3-:----lSi2-t1-7 -8943 314t941'---·------t993~·· 31,181''-

8794 3~54831: 8144 3483253 8804 3114169 8944 3!14j941 8904 3!1?33t3: .. __ 879.! _____ !ll_t,.59_9 __ · _____ .. __ .8$41_ .... ~ .. 3A8 313 .,_. --- . _Ja.0!-'-~----3-Il-48.69 - 89 4!5 3J44J"'J.--------U05'-- - -3813 "/93:

&796 ''''''' ee•• 34&1117 8806 3114t~i 8946 ss4i5!s. eoo6 31"14273: . _ B-'1.2"1 _ 3AtA.:u.t.· ... ~--------8-IA"'.- 34iJIU.' aao1 31! .. oe• 894"1 -3141'1;3. tot? 3!1,404f

8798 345~231. 8148 3481~13 88~8 3!115389 8948 3146113 8908 35'1.13"1!' _ ... 87 99 ___ !4.J1atU_:__ _______ U_4i _____ ,3_4.8U 0!5 -----~ ...... 81.09-- ... 31-1!5,2-J.----- ----1940- - 314' QOJ. ·-- -·-· ------ 10t9--· ----3-81!13-7'1~

e8ö5 1•57399 eaeo 3487765 etöo 3~1644!5 8t!o 354;,,, •ooö 31,6oo'·

Page 79: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside
Page 80: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

Al:........ _______ _N __________________ l~-- ______ _:__}L ... -. U .... J•L -----'l . . N A3.

-~----~------~---------------------~j5( ___ ~-~-~!_ó~_1· ________ _?3~~- ---~"-'!~~~-------- .ll~1:_ --~'r<Ht~~!S .. _____ ,_~Ö1 ........ .38.t.Á88_i _________ I.41.L ... :t8.5Ö48.!~--------t~52 37216! ,. 0102 3751113 0:$52 3'188673 0452 38t,4î., 0412 315012''

~ .. ·-~·-------------------·-··--··-- =~-~~----···1~~:-ft-i;----~,-~~-- . ·--~~;~i~--~·-··········· :~::···----~-~-=~~i~· ···::·-~-~.' '~:1::·:t -::::c ·:::i~:~ 0255 3729105 1 OJO!S· 3'1tQ2~3 '3JI 3~90.305 04Ö5 .li2Ö6U 941!5 38!10!3 ö2'56 .. 3'29441: . '0106 3'161811 ,,,, 3'1907&1 0.06 311!217' 0416 38!5!0~3.

------:~-;~---~~~~1~-r- .. -.-:;~~- -1;11~-:~- -~~ir- -- ~~:i::~---- -::~r ·~-~:t~l~~ .... ---- --- ::::- :::::;r -························--·------· ··-----·-··· --- ---ll~ä ············~-;;;-l~;: ____ --{11~----J~:l~;~- --······-· -~~1~-- ---~~:l!ii·· --·.- ::~:----~==~:::. -···-···:::;-···· ·--~:~!:~;,

... _ ... . ... . .. _____ ,_t~i ~~J~_f,_J_ _ __ ~U-1- .:sl_'":St45 .. lill. 3.113.30! ... 1411. .. 3.1ri:ttt. 0461· 31!5!1613: 9262 313428' 0312 3'163,37 0~-~ 3'1041&9 04f2 312~583 0462 3816201

_ -~-. ---------------------~----------~2-~-~----l?1~2~1_ ____ ~_lS._3 ____ ~l~!8l1~ _______ !lil~-- _U.94.64_t. _______ ,_4.i3_, __ .. :3lf4.8 ö.! . 9.4.6.!..--- .3.8.!562tl'-- _____ _ ..... .. ..... . -- -- --· . 9264 37J448t: 0Î14 376,105 ·~·4· 31950 09 94{4 381635.9 0464 38!J60tl· --~----·---·--·~---------·~-------- ---'-ll1~ ___ 1:?l!J Ó !S 1·. . __ ,_I_U~ ___ .1:l.llll1~------ · 0 S 6 I ....... 3!lt6.tU· _______ ...!W.-- 3.1Ji3it............. 9A6!. ... --3.8U.lU'-·--··- ....

9266 37$6785' 0116 3766005 0366 3'107813 04!6 3816901 0466 38S19St· .. W.?: .... 17..12.UL.~--- 9-tU- ... .3161012 ........... ~.$$i ... 3'.'1-l!t23 . 94i'7. --3'41~4'11 . 0467 PJ.I0-31 9268 37J16d1: 9.18 3168681 OSI8 3197111 0418 38IAOOS· 0468 38589Si'

···----···--------------------------·------~------9..2.6J... ___ !138t.4.SL ______ .....!&J.i. ...... .l.lJJA11.~---------'-Jlt---~UI930S _______ ..9U9. ...... UJ .• J-U.:..-----0469· ·---~!JO!Itl·---0275 3730665 OSAO 3161401 OS'O 3790,37 9410 3110223 041Ö 386110!

.. Jl2.'ti . ... 313.9 665~------· ... til1 ... -..3.1700.7.3 ---- .... .oj;s.,:._ --~.A.ê.O.~-S-------9_.11-..... 3'.8$0-0l$'-- -------l.4-7i·---··"6Îl Ot······ 9272 3140ó97 0822 31?ó601 o'?a 3800.13 04a2 381{161' 9412 316!211· 0.2.7.3 3.:140.865 Oti3 3'111297 O$tl- 31t),12f1 0413 I&IÏS.61 · 0413 31621'1f1

9274 3741753. 0124 317tP17 9$14 3802103 0414 JISiSf'' 9414 386331, . ... 027!1 1~•2113: .... .t.ta.s ---- :U1261l t$-f-8· - :Siq2503 9415 -3-lli:-'O-J>-- - - 0418 3!6168-1'

0276 31417!3: 1!26 317!.33 .,,, 3802809 0416 38338!9 9416 38&41!1: 0271 .... 3741êU' .. 0111 .. 311$.3.3 . --0$Ï1 3Bi31l1 .... -'411 3&3fl'l' 041'1· !161823 92?8 3141985' 0128 311!~33 9318 3104201 0418 3834615 0418 386644, .

. 9.279 _:s141~8.5: .ilt-' 317,ót'1 .. osf• 31Q487.J •• ,,. a•si,it ••'! 31664•1 029Ö 3144081. 0830 3?79077 0310 3805141 04:10 3836131 0410 3866131 1

9_28i - 31-~6óJi1: ..... ià3.1. 3'17&4l7. 03-fl: 38Q6.SU.'7 14!11 -3IIJ11J- 041!!, 31677~\f: 9282 3146''' 0832 3171865 0$82 3806,69 0432 383640!· 0482 3868601' 02.8.3. 3ZJ6.8..4.L..~----~it3-J ...... J.'7.1-1-JI.5 .. - -----1Îf$.-- •-J8j.6J.60-... - ·- -0-4:'13-··· --38J.&11J- - ----c'-48!--- -386109!;• 9284 3'141561: 9334 3111~03 0384 380781'1 0434 3838801· 0484 386030!:

................ 02-S.! ....... !1 . .4JU.i --·-·- . .Q.U,;..._ 3.2.1-lJi-3---· -- .. 0-f-J 3Bg1-l2-t 14D ·383~663 041S · 311Öè1t 028 374026J 0~36 3119111· ,,,. 3809.33 9436 383!663. 0486 387lQSI·

12.4926.5.'.. _____ ita'Z~----4'-ll.7-l!L ...... ___ o,..,.-.~-Ht0-80-t---------U-:~1'.:. ..... .....a .. -o74~------~0ü'1..:.. ____ !1-'1S.G-Il'···· 9288 3'149649· 0838 3180705 O$,f8 38~0119 0438 31.Ï36?' 0488 38,lS10'

-- --· --· ·- ···- ------ ·-·· .... 9.28Cl ____ 3l.,.D.i1.2.~---· 913.9. _____ 37.8.!185. -··· ..... IU-0---~.&t-1-'-1·1- ·-·------04-39----3-8413&-,..:..... 04JCF- - U-1-13tJ' o29ö 3751!'' o#•o 31811&5 v3oo 3St2''' 9440 3&41661· o4oo 3&13óos·

..... 9291 37!2t .• S: 0841 3'82!41 OJOS. !8!211'1 · 9441 38.-.~t.-9!· 0401· 31741 '11 9292 !15!337' 0142 3183~17 0302· 3812,45 0442 384S611 0412 387446:5:

____ 9293. .3 .. 1Jtt.:.u .J?i'3 l'a:uu,7 osJJ .. 31t3113 0443 . 3&4j!-Jt -- 04t-3 31,480ii 9294 3754tS7 1144 3783609 0314 38t428t 0444 3844513 9414 3816231

.. 9.2.95 ....... 315413·1~ 9St4.5 - 31.84!17 ... O;J~!S 3814817 - 0441 -334Jl.i9 - - 0405 38762~0 0296 3'1544'3: •••• 3785001 .,., 3115101 9446 3846863' 9406 3816103: ta2.! .... l2S.It9!: ,,_., P81&73 . oa~' J.l!6tn• 044-'1 38-4t863 04'' 381736'' 9298 3155615· 0848 311&os7· 0308 38168?.3 9448 38•?115 •••a 3811943:

____ ___ ... 9.29Jl.. . .. 31.5.6.8..25 .9.t9 . 3186.205 O;JOt- 31!,6-873 0449 - 38411167 0409 38'116!1· tlöö 3751t6t· os5o 3'181385 04ÖÖ 3817353 94!0 384tttt· osäo 3878611

Page 81: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

~---- ----4~-·· · ·-·N-·· AS.

-------i!fÖL.-_3_~_l_?_21_l 9111 ... _ 3.910043 _9_&fi1 3939419 96!1 3t1t46!5' ·- . .t7C-1: __ 4Ö03711' ----····-·-·-··· osö2 3879863 ~~~~ 39101.87 •ag2 3940479 0612 :n'?i7!J:S 0'62· •ó ö 4Bà$:

-------~'Jll_ 3~1.7986!: '!'3 391.0.6.6.7. t!t13 .. 3941.33.9 t6.1!. 3t,U.S3~ - - -· ''.03- 4óiJ4aöt· O!IÖ4 3880247 OjS4 391!587 06~4 3941729 0614 301~889 97Ö4· 4bÖ!J440 O!JJJ5.. 38.~1!~3: Olf' 30132'9 9615 39.4218!5 961!5 3113811 ''Ó' 40063t:J:

---··-------------- ____ ..... ····-···· ;;_~;_ __ J:;;:it _____ :;:t_ ~-:t::li ....... ::il ~::~:~i- . :::t ~:;Î:IJ'_ . _:;~~ :~~~::: os58 !883dr; ijja Jtif$15 o•öa 30448At 9618 317~289 97óa 4ÖÖ''''

-- ----·-· ---- -- i:~l--- ~l~-:~Il- ·- --:::~- -~:t:!ii- ··--··--:·:I: -~::::~;- ------ :::: ~:;;:ït --- --- :;t~ :~~:;i: 9!'Ü. .l8_S.!!Jt~~- ~1.61 39.i!Oll9 t6i1 3947&21. ,,,,.. :so.;;,,,. 9111 400011~; 051t !88586!. Ot62 3917.83 ••ia 394732i 0662 397~''' 0112 46ót721

------····-·-··------··---·-·-- ·-·-------------------~---«l_!li3 ______ ::5~t-~8J..l, ________ l!.6_3 _______ 39i.1 9.1.5__ ___ _ _ _9fi3. . .... U4 7&21 .... --· -- - - .16'3 .. -- 30.,t41 5 --· ---- --01 13,- - -4Ö! Ö J4f-9514 3887441 ,,,4 3017915 0614 394016t 9664 301~41, 0714 4~1~30··

-----~----~------------------------------~~--•.Jit ____ !_~_._,_ut _____ !'IJ.! _____ 3-ot-tuz ____________ ~•-•.t•~---.ltUJAt------------t-'••- -···----s.o,_.,,!'-----------''-l,.,.-·--4o1tt&t'-''1' !8883Q3. ,,,, 30i0979 06i6 304960t 9666 301{401 ''16 4011553-

___ ,_!Jt;7. ... .ll.l.fUJl...... . .. lU1 - -.39-S.Ut.i: ___ . - 06-i-t -l-OJ-GJ0-1 0661. -308-i-;&J 0111 4G1!U·!S-:-0118 :sea9tOl· •••• 3taóo75 ·~~~ 30S1tll 0668 3t8i2it ,,,. 4g1ss~•·

------------·· -------------------- ~- __________ Q.!J~9-~---3!llUt.:-- ~-iJ6.9..-----Ut.t.8.U --· --0.6!-9-----30-J1-6-4t~--------0-H9--- 3-.fl-&i!MJ:----------0·71--9---~-- 4G1-!S~·-· 952fi !8Ptó87 Ot?O 3022.15 Ottó 3912121 0670 308~5i3 971Ö 401!003 0521----38.91U!: ______ Ot'l-1-- ~0220-!5 --OM-1- 3t-J39tt -- --9--6-'11- 301ttil3: -0,11--- -4Ó110t,. --9522 38923!3 •• ,2 3023675 06t2 3054313 0612 308-8Ö1 072* 4Ö16110 0523 J!i2&!1 tt-!1.3. 30241!15- 96t3 3954~13 96'13 39&.171 9113 4Ö161DI 9524 3893261 ,,,. 3024803 0614 3015073 9614 3088760 0724 4016701' 952! l!U44-6.1.. ''''---3-924103--.. . ---06éJ ---- 3.0!1-5,49 · ~ '-''' -3tt-.6Q9 -· t't5- ·· 4G180!f' 0526 38fSó43· 0'?6 3921513 9116· 39J6729 9676 3986801 0126 4010321·

_ 9.52'1--- .389Sti43 ;'''-- --39.2'!1901 ,.,., 39,714! 06'17 30IJ:t71 9'27 4tl193ti· 9528 389!379 ''78 3926939 olie 30J716t 9678 :so8i4ót t1te 4ö1o:sos. 9520 !.8ifló.S1: . Qj.79 --3.0a.77.1U .. Oftt .JO!Jl4-lt 96'19 3N.4f-t- 9'JII9· 4&2Ö38!: 0535 !897aet· o9ao :s•a'''' •••o 3019777 •••ó :soa1111 o1s5 4~aö83J 0-'3t __ 3~9'!82.1 -~- _9'!J81 .. ~ . .392.92.43.... _IJ3t-- 3tt9717- 06-11- -ltt•-771' ---07-31 · 4021601' 9532 ]897117 0182 3029963 ··~· 306016!5 0682 3096313 0732 402t841:

... O!S.33. __ tl_f..P.zU:. --- _95.13 ..... 3929963--- -~·· -''-•3 · . ---39-60'1-'-- -- --9613 ···39-ti#IJ -9,33~ ····- -482.341 0!534 !000281: ,,.. 3920963 0634 39614!7 9684 3091577' 0734 4023689' 9.5_3J ....... :l~OJU,1: ... 9.1ll .... ~ -.3-'i-3.0.13 .. 1....... . . t6il 306-2Si.3 . 9615 .. l~OjOJ7 9735 4023689 9!536 39004!9 0986 3031147 0636 3962003 9686 39940,3 0736 4Ö24013 _,,_3;_ -~"~ou1s. --····---•-J.t'L ___ J.o.S2o~''---~-~---~•,s-*-·----U'3't1-----···--··•••-7- ... -:soo_.o,a. -- --··--•»1~-- -•ötltt•'·-os:Je !0016t9 owaa 3t:~a•83 ••~• 3964129 9688 39943!3 97!e 4021tti 9.539 __ . -~l,OJl2i~.2 _______ 9_t.89 ..... 3.93!.43 ........... ---- 063-0- . -3064!119---- ---0619 -... -3ttJ.16t.·······-- --- 0'3-9-· ---4-Ö-2&1!,, 0545 39034!9 •-•o 3934883. ·~~~ 3964,tt 06tti 309~3!9 074i 4Ö267'i 9!J4i . 3~0428.3. 0ti1 . 30348!3 .. 06~1 3065173 0601 3.9ft8Q9 0741· 402'1721' 054t 39049S3· oto2 393!!13 06~2 3066141 9602 30tABöo 0742 4ó2824t O!S43 . 39049!3• 0J93 Jl3608:J. 0643 · 30661Ö9 9603 · 309,06-1'- -0743- · 4tt211!4f· 0544 3001339 Ot94 3937213 06~4 306736! 0604 319.873: 9744 4~2844t· 9!54!5 3906381~- o,,, :103.7161 06,11 3068t29 0695- . 3991-813: ,,,, 402001'' 0546 3907935: 0196 3937167 9646 39688Öt 0606 309~0{7' 0'146 4Ö3Ó417' 9547 390!1~1 ••• ., 3031243 .,~,. 396810t 9601 3 ..... ,. ,,,, 4Ó3t06'' 0!148 3908435· •••• 3031723 0648 306892t 0608 4005880 9748 4031281 9!49.. !909371: 9199. 303112-3- 0649 3969903 0609 400{161'-- 0740 4033221· 0555 391D043 oaoo 3931843 06jo 3970703 •16o 4aof8t7 071i 4ö34377

Page 82: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

~3: _ ~----- .Ji __ . .A:L . _____ -~--N. A.!. . .. JL . 43- - N . A3.

9751· 40 34377. . ... _O_~~l~ _ ~06~~6l __ ~-----~~~J~1..:__~_4Jt~~2!t ____ .... . 9.9 Ö1 .. _~t.li~ti!L ______ ~o_iiL. ___ A!.!U.1i.l• ----~----------------·····- - O~f5j _______ f63•7ii: - 01102 4064023 9è'2 40'96201 OOQ2 4~2·8'5 09!12 41!582!19

0753 4ó3B2141: 0103 __ -i0.6.,_811_ ----~'_an-~_-4Jl.2~1!3_____ t9.113 .411.d!!.. _______ .ti.U: ____ AU03l!l' ...... -------------~---------i75•1--4-tf!6ó81: ______ 91oi- 4066003 oaj4 4Q0840t otg4 4ta!•~s· 09!14- 416~04~

.9.!'--' .. 40~~-'"~1: ....... 9JD.5. 4066.6'75 oJ!!. 40t.a_,,,_. . ftn!l 4.tsgo.s.t tt.s.e..• .. 4t•t4t7· 0756 4036561· 0606 4068315 08!6 4098505 9906 4130613 09!16 4161·~-

-----~--------------~-----~--------;~;I---l~l~l~l:-------=~~·----~-~~~~J;--- ---- · --:lil-----!~~l~i;-·-------:lll-----:tl~:~ ------::::-----~t::::!~- -- -------- __________________________ 2.1'.9 __ . 4Ql8iftl; __________ 9_Ui_ _____ 41t!lrt1t· _____ . 9-$,9 .4iQ04ti ..... ____ ot.g9 ... __ 4t:t4t~3 .. tto.t~--4t6.Ut~-- --

0765 403814; oa1o 4070731 '8'0 4100175 9o~o 4133107 oo6o 416~683

-l~:~ :~!ö~:-r -~:i;~~---!lit:~: -:::i :tï:~:; :·:ft ~~~=~ ~::i· :~:;::j: 4040 40 7· ___ --~1_13. ~ __ 40.ltt•~z______ __ .9!.3 .. 4iijUJ,t_ __ __ .t.t~~- ... 4U_f1;1~---- . toa:s: - 4St66t.Si.; .... -----

97 6, :~ :i:~i- ::i;_ :t;~i:: ________ -~t!: _______ :t:I::; __________ ;.:r:~-----:~:::1r ___ ----- ::::~ __ :t:t~il__ _ _______ . ___ _ -------·----·-----· -------·----------------------------;,6~--------4o-432ei:· -- ---öit6 ·4o7:SS39 oè66 41Ö54&3 oo16 4t36oas oo&6 4t6&48S, .

.. 9'l_~_,_ 4Jl~l211. . ... 9f17 ... 4Jl.'l!1U........... 98,f7 41Q516.7 99f7 ·-~3;11? ,,,, 4!68483:. 9768 4041665· t111 407!101 oè'a 4tb60S5 ooi8 41!~387 9968 4!68483 4 . . . .. ' .. .. . . ~- ..

----------------~-~~~-------~~{!-~ ;;----- ~:-H~---· --:~l~:il-----------:ttt~---:ig::~~-- -----···· -::;:--- ··· :t::·:!:---- -::~l :~ ::~-g~--____ iz_zi ____ A.D4_~~J.1: _______ 1tt1 4o7.7~''------------'-'f-1----•tösJ?t_ .o.ot1 .4t3-~2ol~-- ____ o,_,,_ 4t 'o.ai! ·

0772 4046ó65_ •• 22 4078115 9872 4108723 0912 4!30827' 9972 4171171 9.11! 40.4.6917 oaa3 4071a,05 oa;s. •s.toos.t tt-13 4t45427 o.073 417!,11 0774 4ó48311: tl24· 4078603 oaf• 4iio•o' 09~4 41•io~t 0974 4!7!16!

_ .. _971.J ___ .4J1Üll1· __ tJU-----4-C'?-1611· o.t~------4!-~'14~--------- . - 0015-- ----4~-4Ï!IJ.fl-, .. --- --------0.0-~- -•t1!76f,·· 9776 4048S61. 0816 4080475 o&;6 41i217t 0916 4141809 0976 4!74003:

---·-- ----------------------- -----~------ --------------------· - .... 97 .. 7'J. -- .. Atl-4.9t.$.t.; .... ________ .J-lt7 --···----4.0..8-t,-lA-7 ··· -- -- - - -0Jf7' --- 4~t»J-f.---------- 0~1-'1- 4! .. 1t 9 ···· -- -- --0-911 -- -4-t"' lf5;IL 0778 40499ö5: 0128 4081!!111 08!1 4't3523 0918 4t4~t87 09?8 417!631 9179 40'-1.0!7' .. ..lUS! 44J&~-n3! •• ,, -41t!f't3 0.919 --4!l:tlf0tt -09'19 -41?6!1.1' 978Ö 4011441: OA!O 408!235 98AO 41!3107 0930 414,779 OOIÖ 4!76111'

_,_781- 40l2ti4.t: ____ o.a.u ...... Aoau:ss Olf1 •il-5.2!!1-t-----------0.0-1-t.c ___ _.l••~~t o•ai 41"7-79-!IL 9782 40$2905' 9&32 408!.!!11 0182. 4tt63!t 0932 4t4,8tt 9912 4t?IIOO-

---·------- _91Jt1. ___ 40,29H_!~ __ __ __ 9$JI3. ..... 4.08.4.6.9t____ . t$1! -4it6,!9. .. ---0-03-3 ... -4!4;!17 098! - -4t,I21J0-·--0784 4053241: Oè34 408!6!!19 9é~4 4!16739 0934 4i4i8t9· 0984 41?810P

________ ll'l.B.l.. 4.0.f4681: . 9.1l! 40.4.6.0A.3~- OÜJ .. 4l174lt -0031 ···· - 4!.4,1!9 908!J· 4!79067 978~ 40~!J?3? 0836 4086499 OBÀ6 4118083 9936 4t4Á203 9986 4!70831·

_____ 9787~- ... 40'~ àö1· _ .. oa:u __ 4o8tot.9 ----- ot-a? -- 41t8t83 - - - t-os1- ---4-&4tt-3t .... oot? 4t8Ö4IO 0788 40S6o01 9138 4087723 oe~s 4118227 oo~a 4(4~219 ooaa 418Ö79S 078o .. 4051345 ~a tso 4o&77_1_3~---···-- .. t.eao. ..411'''' ooso ____ •~tto.tst: o9eo- 4181Dt1·

·-cf79ö --·--4o'79ii: ta4o 4087723 t8tà 4tao723 oo•ó 4t!JÖ2!9 090Ö 4181810 0!91 4Q?~-!~1· 0141 40.887'1.~. .. 08.11 41211143 0941 415Ï81'?. Otl1 41.81810· 9792 40,8065• 0142 4Ó90D2? 9802 41.21683 0942 4t113Ó'1' 9902 4183330• t793 40'8'14' __ oa4s 4óto:u.' _o_u:~_ .Jta.a••• Oll43 4!t!li3ö'? ___ 9oo:~ 4183963: t'194 ·· 4o-SJ793 ____ -- -,844 4090747 oao4- 412355' oo•• 4tJio7o oot4 418!106'?'

... ___ fl7~9_!L --~Qtgz._!l~----- ___ lá45 _4Ö 92283 _ _ __ OAO.! ... : .. 4i231!5 0945 -4t!J3911 - ------0-9.05 - 41860!'1' 0796 4060713- 0146 409!219 0806 41244!t 9046 4t!Si061 0906 41.864!19 .919.1. . 40l'-521: .. _ .tl.41 40 OS219 01.0'1 4125.t.tt'1 094-7 4tSIJ8SI 9007 4186801 9198 4062481: 0848 4093&03 0808 41258!1. 9948 41.1~83! 9908 41.8'1707'

- 9.799_ 4flJ2.48.1: ...... ..Jtü ....... 4.ó-OU1-1- . 08.0.9-- ---41~48-3!--··--····· 9949 -- 4!16415 .... 0109----4181'1-ol:~. 0855 4063237 oa'o 409!211 t9öo 4t268tt 09!o 4it*1il tooö& 41818,,.

--------- ........... --------·----~--~-----------------

Page 83: Number of lattice points on the surface of a sphere of ... · Number of lattice points on the surface of a sphere of radius Sqrt[N] (N=1 (1) 10000 . Number of lattice points inside

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