Yarosh Gen Fermat's Comments

1

V.S.Yarosh

The state unitary enterprise All-russian research institute For optical and physical measurements (sue VNIIOFI)

RUSSIA,119361, Moscow,Ozernaya,46. [email protected]

Non-modular elliptical curves as established fact and

as result application Abel group and Diophantine equations for solutions

Fermats Last Theorem and Conjectures: Riemanns, Beals, Birchs and

Swinnerton-Dyers. (Survay problems)

Abstract If whole numbers v and u are such that v > u and

greatest common divisor GCD (v, u)=1 , at that v and u of different evenness, than triads ooo c,b,a , generate a endless series of dyophantines

equations:

220

0

220

uvcuv2buva

+==

=

as endless series a primitive solutions for Pythagorean equations: 2o

2o

2o cba =+

and as endless series irrational roots:

n n222n2222n22222

n no

2no

2o

2no

2o

n 2n222n2n222

n 2no

2o

no

2n2o

n 2n222222n22n22

n 2no

2o

2no

2o

no

3/])uv()uv2()uv()uv()uv[(

3/)cbcac(c

3/])uv()uv2()uv2()uv()uv2[(

3/)cbbab(b

3/])uv()uv()uv2()uv()uv[(

3/)cabaa(a

+++++==++=

+++==++=

++++==++=

for Fermats equations : nnn cba =+

With all this going on, triads ooo c,b,a generate :

1. A five forms

2

for separation endless series prime numbers, see below Part 2:

05

00004

00003

002

02

02

002

02

01

cAbaif)ba(Aabif)ab(A

baif)ba(A

abif)ab(A

=>=>=>=>=

2. A non-modular elliptical curves. For this curves is special variant Freys-Yaroshs equations:

2o

2o

2o

2

cba

)BX(X)AX(Y

==+=

and general variant equations:

no

no

no

2n

cba

)BX(X)AX(Y

==+=

where

no

2o

no

2o

aBoraB

bXorbX

====

3. A orthogonal coordinates for complex numbers:

oo

oo

biasbias=+=

and for complex function (complex invariant):

2222o

2o

2o

oooo

)uv(cba

)bia()bia(ssS

+==+==+==

4. A two zeta functions: Riemanns zeta function

0)S1()S1(2Ssin2)S( 1SS =

and autors zeta function:

==n

SQsin2)1()S( s

a key to finally proofs a conjectures: of Riemanns, of Birch and Swinnerton-Dyers

5. A statement:

Hypothesis of Shimura-Taniyama All elliptic curves are modular curve,[1],

is erroneous hypothesis . According, reasoning by doctor A.Wiles, [2] , is faulty reasoning.

As alternative and as confirmatory evidence this fact, autor to make an offer application Abel group and

Diophantine equations as universal mathematical formulation for proofs

3

Fermats Last Theorem and Conjectures: Riemanns, Beals, Birchs and Swinnerton-Dyers.

PACS numbers: 02.10.Ab 02.30.Xx

T a b l e o f c o n t e n t s

Instead of Introduction Part 1, Yuri Zhivotov , see [5],

against argumentations by Ken Ribet and Andrew Wiles . A u t o rs P r o p o s a l s Part 2 , History

Part 3 , Possible variants proof of Fermats Last Theorem over Q and applications Abel group. Alternative Wiles proof

Part 4, Non-modular elliptic curves, 10-th problem of D.Hilbert, as way to proof of Riemann Hypothesis

Part 5 , Intercommunication between the elliptical curves, Abel group and the non-modular forms.

Part 6, Riemanns sphere as mapping of space task common solutions of Conjecture

Birch and Swinnerton-Dyer and Riemann Hypothesis

Part 7 , Proof Riemanns Hypotesis Part 8 ,C o m m e n t a r y

to the question of Dyophantines equations and their irrational roots ,containing

information of non-modular elliptical curves. Part 9, Proof of Conjecture Beal Reference

4

Instead of Introduction

SYSTEMS A COORDINATS AND GENERAL INVARIANT

System A coordinats for whole primitive Pythagoras numbers

Pic.1

This is ortogonal system coordinate for primitive

Pythagorean triads:

22

0

0

220

uvcvu2b

uva

+==

=

as for all pair v>u numbers are the numbers

of various evenness taken from endless series , see (3) :

,...4,3,2,1)1u(v,....,4,3,2,1,0u

=+==

5

C o n s e q u e n c e 1

2o

o2o

2o

oo

2

ucu2

buav

vcv2

bavu

==+=

=== (4)

System B coordinates for my complex numbers

Pic. 2

Geometrical interpretation a vectorial product

Pic.3

6

Here we create a product:

220

20

0000

ba

)bia()bia(ssS

=+==+==

(5)

It is general common mathematical invariant for systems A and B AS COMPLEX FUNCTION

over field N of natural numbers uv >

Other, we create a vectorial product:

[ ]s,s (6) and vectors length :

Qsinsss,sS == (7) as argument for equations:

0)S1()S1(2Ssin2)S( 1SS == (8)

GENERAL MATHEMATICAL STRATEGICS A SOLUTION OF THE BIRCH AND SWINNERTON-DYER CONJECTURE

Defining role in the proof by Birch and Swinnerton-Dyer belongs to

the EXPANDED or CLOSED plane of complex numbers: += iz and = iz

properties of this plane are defined by properties of Riemanns Sphere below, on Pic.4 Riemanns sphere [3] is represented.

In a Fig. 5 it is presented diametrical section of Riemanns sphere

Here : Flatness ),( for complex numbers += iz and = i'z ;

Point )b,a(PP oo= ; Point )Y,,('P'P = ; Axis absciss for ; Axis ordinat for ;

It is stereographycal projection flatness ),( to sphere )Y,,(

Pic. 4

7

Stereographycal projection keep angles . thats have mathematical description, see (32):

2o

2o

2o

2o

2o

2o

2o

o2

o2

o

o

20

02

o2

0

o

c1c

ba1bay

c1b

ba1b

c1a

ba1a

+=+++=

+=++=+=++=

In a Fig. 5 it is presented diametrical section of Riemans sphere

Pic.5

INITIAL DATA

V A R I A N T D A T A 1

If angle =Q , see Pic.2 and Pic.3, then:

0)s(Re0)s(Re

ibib

o

o

==++

(9)

According we have endless series numbers:

2o

oooo

ibaibassS

==+==

(10)

8

as endless series primitive numbers of Pythagora over field N natural numbers )uv( > :

)uv(S 22o +== (11) In the end we have endless products,

as endless series seros:

00sinsss,sS === (12) According this statement, we create endless series

functions for proof a Birch and Swinnerton-Dyer conjecture

0)S1()S1(2Ssin2)S( 1SS == (13)

Here scalars 0S = are arguments for theyre functions .

Pic. 6

The axis of absciss oa is geometrical axis symmetry for two a system coordinat.

Common the angle Q is general argument for general functions: 0)S1()S1(

2Ssin2)S( 1SS == (14)


If angle 0Q = , see Pic. 3, then:

9

o

o

o

o

a)s(Rea)s(Re

0ib0ib

==

==+

(15)


20

0000

a

)bia()bia(ssS

==+==

(16)

as endless series primitive numbers of Pythagora over field N natural numbers:

)uv(Sa 22o == (17) In the end we have endless vectorials products, as endless series sero:

00sinss]s,s[S === (18) According this statement, we create endless series functions

for proof a Birch and Swinnerton-Dyer conjecture:

0)S1()S1(2Ssin2)S( 1SS == (19)

Here scalars 0S = are arguments for theyre equations.

0bi+

oib Pic. 7

10

Arithmetic of modular elliptic curves

and diophantine equations

Basis of arithmetic of modular elliptic curves contains five statements.

Statement 1

Let 1N whole number, and let )N(0 is multitude all matrix:

2221

1211

aaaa

)22(M = Where )a;a;a;a( 22211211 whole numbers , N divide 21a and :

1aaaa

)22(DDet

12212211

===

=

Statement 2

(Mazur)

A Frey elliptic curve: )BX(X)AX(Y2 +=

has no Q-rational subgroup of prime order 2n The inexistence of Q-rational points of prime

order 2n on Frey curves is sufficiently.

Statement 3

Hypothesis of Shymura-Taniyama All elliptic curves are modular curve,[1]

is sufficiently

Statement 4

A.Wiles proved: Hypothesis of Shymura-Taniyama equitable for a Frey elliptic curve.

Statement 5

If Fermats Last Theorem is proved for n=4, there is no need

of prove it for all even exponents of degree for Fermat equation Autors [3] and [4] it is claimed that :

Suffice it to prove for n=4 and for n=p . Here p arbitrary value of prime numbers.

11

Arithmetic of non-modular elliptic curves and diophantine equations

Basis of arithmetic of non-modular elliptic curves

contains five counterstatement.

Counterstatement 1

Let 1N whole number, and let )N( is multitude all matrix:

333231

232221

131211

aaaaaaaaa

)33(M =

where

)a;a;a;a;a;a;a;a;a( 333231232221131211

whole numbers , and :

0)33(MDet =

Equivalent matrix )33(M :

===

===

===

=

no33

2no

2o32

2no

2o31

2no

2o23

no22

2no

2o21

2no

2o13

2no

2o12

no11

cabcaaca

cbabaaba

caabaaaa

)33(M

Matrix )33(M based at the primitive triads of Pythagorean.

Key condition for primitive Pythagorean triplet. Diophantus and then Fibonacci , indicated the following method

of search of solutions of Pythagorean equation :

If whole numbers v and u are such that v > u and greatest common divisor GCD (v, u)=1 , at that v and u

of different evenness, than triads ooo c,b,a , generate a endless series of diophantines equations:

12

220

0

220

uvcuv2buva

+==

=

as endless series a primitive solutions for Pythagorean equations:

2o

2o

2o cba =+

and as endless series irrational roots:

n n222n2222n22222

n no

2no

2o

2no

2o

n 2n222n2n222

n 2no

2o

no

2n2o

n 2n222222n22n22

n 2no

2o

2no

2o

no

3/])uv()uv2()uv()uv()uv[(

3/)cbcac(c

3/])uv()uv2()uv2()uv()uv2[(

3/)cbbab(b

3/])uv()uv()uv2()uv()uv[(

3/)cabaa(a

+++++==++=

+++==++=

++++==++=

for Fermats equations : nnn cba =+

with all this going on, triads ooo c,b,a generate :

1. A five forms for separation endless series prime numbers,

see below Part 2:

05

00004

00003

002

02

02

002

02

01


baif)ba(A

abif)ab(A

=>=>=>=>=

2. A non-modular elliptical curves. For this curves is special variant Freys-Yaroshs equations:

2o

2o

2o

2

cba

)BX(X)AX(Y

==+=

and general variant equations:

no

no

no

2n

cba

)BX(X)AX(Y

==+=

where

13

no

2o

no

2o

aBoraB

bXorbX

====

3. A orthogonal coordinates for complex numbers:

oo

oo

biasbias=+=

and for complex function (complex invariant):

2222o

2o

2o

oooo

)uv(cba

)bia()bia(ssS

+==+==+==

4. A two zeta functions: Riemanns zeta function

0)S1()S1(2Ssin2)S( 1SS =

and autors zeta function:

==n

SQsin2)1()S( s

a key to finally proofs a conjectures: of Riemanns, of Birch and Swinnerton-Dyers

5. A statement:


is erroneous hypothesis . According, reasoning by doctor A.Wiles, [2] , is faulty reasoning.

As alternative and as confirmatory evidence this fact, autor to make an offer application Abel group and

Diophantine equations as universal mathematical formulation for proofs

Fermats Last Theorem and Conjectures: Riemanns, Beals, Birchs and Swinnerton-Dyers.

Here pair natural numbers

uv > as basis for solutions common Problems

o f F e r m a t , Beal , R i e m a n n , B i r c h a n d S w i n n e r t o n D y e r

Whole numbers ooo c,b,a is abelian varieties over Q

and over field natural numbers uv >

14

Counterstatement 2

A Frey elliptic curve : )BX(X)AX(Y2 +=

has Q-rational subgroup of prime order 2n The existence of Q-rational points of prime order 2n on Frey curves is sufficiently.

Counterstatement 3

Hypothesis of Shymura-Taniyama All elliptic curves are modular curve,

is faulty Hypothesis.

Counterstatement 4

A.Wiles proved: Hypothesis of Shymura-Taniyama equitable for a Frey elliptic curve

as semistable modular elliptic curves. It is sufficiently. But there is just one snag (to it):

Exist my non-modular elliptic curve. For this curve is special variant Freys-Yarosh equations:

2o

2o

2o

2

cba

)BX(X)AX(Y

==+=

and general variant equations

no

no

no

2n

cba

)BX(X)AX(Y

==+=

where

no

2o

no

2o

aBoraB

bXorbX

====

and )ab(Aor)ab(A no

no

2o

2o == conductor-controller.

Instead a formula for selection prime numbers, see [1]:

pp 1pa += where p is prime numbers and p

is analog for numbers:

+=

)4(mod1pfor1p)4(mod1pfor1p

Np

15

I propose three Rule for computations all prime numbers iA

Because:

Exist five forms

for separation endless series prime numbers, see below Part 2:

05

00004

00003

002

02

02

002

02

01


baif)ba(A

abif)ab(A

=>=>=>=>=

Here every number from endless series prime numbers

)ab(A 202

01 = if oo ab > will not divide into

222

22

0

0

)()2(

)122b(4)vu2b(416

=======

If number

2

n4ooo )cba(16

=

is function a discriminant my elliptic curves:

8

n2ooo

2)cba( =

At the same time )ab(A 2o

2o = will not divide

3

22o

220

3)12a(9

)uva(927

====

===

if odd number 27 is equivalent:

+= 2

o

3o

ba427

where )b27a4( 2o

3o +=

discriminant for canonical form any elliptic curves:

oo32 bxaxY ++=

16

Endless series prime numbers:

002

02

01 abif)ab(A >= is endless series of conductors-criterions

for separation non- modular elliptic curves. Number 2 end number 3 were invariants

from endless series prime numbers. All prime numbers settle down in a natural line in pairs,

intervals between which submit to a rhythm of numbers v=2 and 3a0 = :

2

2

2131721113

2711257235123112

=======

2

2

3

2

243472414323741

323137223312192321719

===

====

27173263712616325761253572515324751

2

2

2

2

=======

2

2

2

2

2

2971012939728993287892838727983

327379

======

=

......................2117121211311721111132109111210710921031072101103

2

2

2

=======

All differences between the next simple numbers are subordinated to the law of formation of primes-numbers and spectral invariant:

2)2...2...2222( 32102 +++++++=

Following spectral forms contains this invariant:

21ii21ii )AA(or)AA( == 21ii 2)AA( = 21ii 3)AA( = 21ii 4)AA( =

17

Note: AandA 1

are isogenous abelian varieties

Counterstatement 5

If Fermats Last Theorem is proved for n=4, there is no need of prove it for all even exponents of degree for Fermat equation

Autors [3 ] and [4] it is claimed that : Suffice it to prove for n=4 and for n=p . Here p arbitrary value of prime numbers.

It is faulty principle.

Explanatory example for Counterstatement 5

Initial data

If 4n = , then common multiplier:

3503/)cba(D 2n0

2n0

2n0n

4 =++= If 8n = , then common multiplier :

3450203/)cba(D 2n0

2n0

2n0n

8 =++= Here primitive Pythagorean triplet:

5uvc4uv2b3uva

220

0

220

=+=====

and basis

1u2v

==

Statement:

If equation:

444 cba =+ have roots:

...511801.4c...041031.4b...996355.3a

4

4

4

===

then roots:

18

...507634.4c...2629624.4b...9671335.3a

8

8

8

===

for equation

888 cba =+ is independent roots .

Because:

...]507634.4c[]...)511801.4()c[(...]2629624.4b[]...)041031.4()b[(...]9671335.3a[]...)996355.3()a[(

8224

8224

8224

======

With all this gong on two triads irrational value roots For Fermats equation:

nnn cba =+

First triad for degree 4n = :

...499635512.33

1503

509Daa 44n n42

04 ====

...041031009.43

8003

5016Dbb 44n n42

04 ====

...518010018.43

12503

5025Dcc 44n n42

04 ====

Second triad for degree 8n = :

...96713355.33

1840503

204509Daa 888 n82

08 ====

...262962429.43

3272003

2045016Dbb 888 n82

08 ====

...507533969.43

5112503

2045025Dcc 888 n82

08 ====

19

GENERAL RESULT:

...]507533969.4c[]...)518010018.4()c[(...]262962429.4b[]...)041031009.4()b[(

...]96713355.3a[]...)499635512.3()a[(

8224

8224

8224

======

C O M M E N T

Two the way work out a problem of P.Fermat: 1. Deductive (intuitive) way

2. Inductive way

1. Deductive way

Let 888 zyx =+ equation of P. Fermat A priori it is known:

...507533969,4z...262962429.4y...967133355.3x

===

Issue:

How did I do it ?

Answer:

Enigma

3. Inductive way

Let Cz,By,Ax nn === whole or rational numbers. Then:

n

n

n

cz

By

Ax

===

roots for equation of P.Fermat: nnn zyx =+

If CBA =+

where vectors

20

)cba(cDcC

)cba(bDbB

)cba(aDaA

2no

2no

2no

2oN

2o

2no

2no

2n2ob

2o

2no

2no

2no

2on

2o

++==++==++==

composed of primitive Pythagorean triplets.

220

0

220

uvcuv2buva

+==

=

4. Basis for Inductive way is Abelian group of three whole numbers:

cCvectorsorccsquares

bBvectorsorbbsquares

aAvectorsoraasquares

2o

2o

2o

Abelian group of whole numbers a,b,c (from Internet)

I chose to begin with the notes out of which I constructed the central definition below. The equation which defines distributivity is:

a(b+c) = ab + ac

This has, of course, a `reversed' form, (b+c)a = ba+ca: I chose to name the displayed form `left' distributive and this latter form `right' distributive. When cast in the general terms of binary operators, naming multiplication f and addition g, we have, for any legitimate a, b

and c:

f(a, g(b,c)) = g(f(a,b), f(a,c))

Thus, if we take (AB|f:C) and (DE|g:F) as temporary namings for the domains and ranges of our binary operators, we obtain

a is in A; g(b,c) (in F), b and c are in B; f(a,b) (in C) and b are in D; and f(a,c) (in C) and c are in E.

so we need F to be a subset of B and C to be a subset of D and of E. I chose to take C=D=E=F=B for this left-distributive case, replacing B with A for right-distributive.

21

Distributivity

A binary operator, (AB|f:B), left-distributes over a uniform binary operator, g, on B precisely if, for every a in A and b, c in B: f(a,g(b,c)) = g(f(a,b),f(a,c)). We say (BA|f:B) right-distributes over (BB|g:B) precisely if, for every a in A and b, c in B: f(g(b,c),a) =

g(f(b,a),f(c,a)). One binary operator is said to distribute over another precisely if the former both left-distributes and right-distributes over the latter - in which case both are necessarily

uniform and the two are parallel (that is, they act on the same space).

In particular, any Abelian binary operator which left- or right-distributes over some binary operator inevitably distributes over the latter. When B and A are distinct, (AB|f:B) can

only distribute from the left over anything, and that must be over some (BB|:B), so there is no ambiguity in refering to such an f as distributing over some g, implicitly uniform on B. It should also be noted that if f does left-distribute over some g, then its transpose, (BA| (b,a)-

>f(a,b) :B), right-distributes over g.

Further reading

An (AB|:B) may left-distribute over a (BB|:B): compare and contrast with an (AA|:|) left-associating over an (AB|:B). The combination of these forms the cornerstone of the

notion of linearity, which underlies such fundamental tools as scalars and vectors.

P A R T 1

Yuri Zhivotov , see [5], against

argumentations by Ken Ribet and Andrew Wiles.

A u t o r s P r o p o s a l s

Where is the Logic of Great Fermat's Theorem Proof?

The analysis made shows that one should not believe Singh's book. This is a literary

work. So, there should be a General proof of the Great Fermat's theorem. There is not any. Some people say that Frey established a link between the Fermat's theorem and Taniyama-

Shimura's hypothesis. The others assert that Frey assumed that the proof of Taniyama-Shimura's hypothesis would automatically prove the Great Fermat's theorem. But Frey's article is inaccessible for a reader. The third assert that Ribet proved Frey's assumption.

The forth consider that Ribet proved that Frey's curve was not modular. The fifth consider that Taniyama-Shimura's hypothesis was proved by Wiles, and so on. Moreover, the

assertions of some people contradict to the assertions of the others. Everybody admires the proof of Great Fermat's theorem but nobody saw it in full scope. And taking into account the proofs of Fermat's theorem for the cases n=3, n=4. The riddle of a proof. The search in

the Internet did not help to solve the riddle. The result is very unexpected and pitiful. However there are three personages in this riddle: Gerhard Frey, Ken Ribet, Andrew

Wiles. Evidently, one should look into the works of these mathematicians. Perhaps then it

22

will be possible to understand how the Great Fermat's theorem was proved. Let us believe that none of the mathematicians, except Frey, made errors (Singh informs on his errors). If

any suspicions arise that additional errors exist, we will substantiate them.

The main mistake of Andrew Wiles it the fact that he got involved with the proof of Great Fermat's theorem. All the more, the mathematicians should not have made a mistake

collectively. Let us trace the way from Taniyama-Shimura's hypothesis

to Fermat's theorem, of course, if we manage it. Taniyama-Shimura's hypothesis states that any elliptical curve is modular.

In particular, the elliptical curve described by the equation: )DX()KX(XY2 += (1)

with the integer coefficients must be modular. Andrew Wiles proved Taniyama-Shimura's hypothesis.

That is he proved that the elliptical curve described by the equation: )DX()KX(XY2 += (1)

with the integer coefficients was modular. That is the following equations correspond to modular curves:

)5X()3X(XY2 += (2)

)25X()9X(XY2 += (3)

)125()27X(XY2 = (4)

)625X()81X(XY2 += (5)

)3125X()243X(XY2 += (6)

Equation (6) may be written in the form:

)5X()3X(XY 552 += (7)

Or )BX()AX(XY nn2 += (8)

Ken Ribet proved that the elliptical curve

described by the equation

)BX()AX(XY nn2 += (9)

was not modular. Ken Ribet contradicts to Andrew Wiles. This is a deadlock. Perhaps, somebody wants to say that the numbers nA and nB do not exist?

Perhaps, somebody wants to say that the numbers nA and nB are included into the hypothetical Fermat's equation

nnn CBA =+ (10)

23

and therefore the numbers nA and nB do not exist?

Everybody has the right to make assumptions but assumptions must be proved.

Besides, it should be taken into account that in the equations (2)-(6) also have an uncertainty.

The equations can be put into another form (in an analogous way equation (6) was modified into equation (7)).

But the theory of elliptical curves does not know how to determine the discriminant of the equations.

It is strange that the discriminant can be found for equation (7).

3.8 At one of the steps of his proof Ribet proposes to pay attention to the minimal discriminant of Frey's curve (as it appeared, existent, semistable and modular).

Ribet proposes to write the minimal discriminant of Frey's curve in the following form:

8n2 2/)cba(d = (11)

Pay attention. If number "c" does not exist, the discriminant does not exist. If the discriminant does not exist, the Frey's curve does not exist.

If the Frey's curve does not exist, the curve is fictitious. Just another time Ribet tries to divert a reader's attention

from the real existence of Frey's curve. Let us remind a reader that the discriminant of elliptic curve has the form:

2nnnn )]ba(ba[16D += (12)

Therefore, the minimal discriminant can be written in the form:

82nnnn 2/)]ba(ba[d += (13)

or 82nn 2/)]C(ba[d = (14)

Such record of the minimal discriminant excludes the doubt about existence of Frey's

curve, which is semistable and modular. Such record of the minimal discriminant reduces the idea of Fermat's theorem proof to the consideration of possibility of minimal

discriminant factoring, that is number c presentation in the form of exponential number nc . In this case the Ribet's exercises with putting the Fermat's numbers

into the Frey's elliptic curve equation are not needed. However, as it follows from the article, Ribet made every effort to conceal this

simple truth and to substitute it with the reasonings about the link between Frey's curve and Fermat's equation.

Let us consider the reasons, for which Ribet forcedly conceals the truth.

24

Conclusions

Ribet did not prove Fermat's theorem in the assumption of the truth of Taniyama-Shimura hypothesis.

Ribet made too many mistakes and discrepancies, which allows to consider his proof as an unsuccessful attempt.

The existence of Bill's conjecture also strikes a blow at Ribet's proof, which obviously is impossible to ward off.

Let me express perplexity to Andrew Wiles, which is connected with the use of Ribet's work in the "general" proof of Fermat's theorem,

to which Wiles has pretensions. So many mistakes were revealed in several lines of the proof that it is hard to believe that they had not

been noticed by the specialist in this field of mathematics. The Great Fermat's theorem is connected with

the history of mathematics, and it is impermissible to treat it haughtily.

A u t o rs P r o p o s a l s

Autor of these article proposes to write the minimal discriminant

of Frey's-Yaroshs non-modular elliptic curve:

2o

2o

2o

2

cba

)BX(X)AX(Y

==+=

(15)

where

2

o

2o

aB

bX

==

(16)

And )ab(A 2o

2o = (17)

in the following form:

02

)]uv()uv2[(2

]uv()uv2()uv[(2

)cba(

8

n244

8

n22222

8

n2ooomin

=

+=

==

(18)

If v > u are the numbers of various evenness taken from endless series of natural numbers , vide supra, then :

25

n

o2n

o2

o2n

o2

on

2no

2o

no

2no

2o

n

2no

2o

2no

2o

no

n

cbcacC

cbbabB

cabaaA

++=++=++=

(19)

three vectors over field Q natural numbers.

It is objects to Abel group f,G with binary dealership: Action associative

oooooo

oooooo

ooooo0

b)ac(baca)cb(acbc)ba(cba

===

(20)

Left distributive

ooooooo caba)cb(a +=+ (21) Right distributive

ooooooo acaba)cb( +=+ (22) The combination of this form the cornerstone of the motion of linearity,

which underlies such fundamental tools as scalars and vectors

n

o2n

o2

o2n

o2

on

2no

2o

no

2no

2o

n

2no

2o

2no

2o

no

n

cbcacC

cbbabB

cabaaA

++=++=++=

(23)

With all this going on:

n

on1

o

no

n1o

no

n1o

c)c(

b)b(

a)a(

===

(24)

1Ecba 0o0

o0

o ==== (25)

mn

om

on

o

mno

mo

no

mno

mo

no

ccc

bbb

aaa

+

+

+

===

(26)

n

mno

mno

mno D3)cba( =++ (27)

where

26

3/)cba(D mnomn

omn

on ++= (28)

Consequence

Linear combination

n n

o2n

o2

o2n

o2

on n

n 2no

2o

no

2n2o

n n

n 2no

2o

2no

2o

no

n n

3/)cbcac(CCc

3/)cbbab(BBb

3/)cabaa(AAa

++===++===++===

(29)

are endless series roots to Fermats equations :

nnn CBA =+ (30)

Here and everywhere:

22

0

0

220

uvcuv2b

uva

+==

= (31)

primitive Pythagorean triplets

I did consider all cases that appear at the "introduction" of the numbers

nA ,

nB and nC (32)

from Fermat's equation:

nnn CBA =+ (33)

into general the Frays-Yarosh

non-modular elliptic curve equation:

no

no

no

2

cba

)BX(X)AX(Y

==+= (34

if vectors

27

nno

2no

2o

2no

2o

n

n2no

2o

no

2no

2o

n

n2no

2o

2no

2o

no

n

CcbcacC

BcbbabB

AcabaaA

++=++=++=

(35)

generate whole numbers as linear combinations

)bcac(Cc

)cbab(Bb

)caba(Aa

2no

2o

2no

2o

nno

2no

2o

2no

2o

nno

2no

2o

2no

2o

nn

+=+=+=

(36)

In this case

linear combination generate three multitude irrational numbers

n n

o2n

o2

o2n

o2

on n

n 2no

2o

no

2n2o

n n

n 2no

2o

2no

2o

no

n n

3/)cbcac(CCc

3/)cbbab(BBb

3/)cabaa(AAa

++===++===++===

(37)

as endless series odd numbers or as roots to Fermat equation:

nnn CBA =+ (38)

Pay attention.

If whole primitive Pythagorean numbers

220

0

220

uvcuv2b

uva

+==

= (39)

does exist, then the discriminant

02

)]uv()uv2[(2

]uv()uv2()uv[(2

)cba(

8

n244

8

n22222

8

n2ooomin

=

+=

==

(40)

also does exist.

28

If the discriminant does exist, then the Frey's Yaroshs

non-modular elliptic curve also does exist.

D E D U C I N G


is erroneous hypothesis . According, case- based reasoning by doctor A.Wiles, [3] , is erroneous either.

P A R T 2 History

In mathematics, the modularity theorem establishes an important connection, between elliptic curves over the field of rational numbers and modular forms, certain analytic

functions introduced in 19th century mathematics. It was proved, for all elliptic curves over the rationals whose conductor (see definition below) was not a multiple of 27, in

fundamental work of Andrew Wiles and Richard Taylor. The result had previously been called the TaniyamaShimuraWeil conjecture, or related names. The great interest in the

theorem was that it was already known to imply Fermat's Last Theorem, a celebrated unsolved problem on diophantine equations.

The remaining cases of the modularity theorem (of elliptic curve not with semistable reduction) were subsequently settled by Christophe Breuil, Brian Conrad, Fred Diamond,

and Richard Taylor .

An incorrect version of this theorem was first conjectured by Yutaka Taniyama in September 1955. With Goro Shimura he improved its rigor until 1957. Taniyama died in 1958. The conjecture was rediscovered by Andr Weil in 1967, who showed that it would follow from the (conjectured) functional equations for some twisted L-series of the elliptic curve; this was the first serious evidence that the conjecture might be true. In the 1970s it became associated with the Langlands program of unifying conjectures in mathematics.

It attracted considerable interest in the 1980s when Gerhard Frey suggested that the TaniyamaShimuraWeil conjecture implies Fermat's last theorem. He did this by

attempting to show that any counterexample to Fermat's last theorem would give rise to a non-modular elliptic curve. Ken Ribet later proved this result. In 1995, Andrew Wiles, with

the partial help of Richard Taylor, proved the modularity theorem for semistable elliptic curves, which was strong enough to yield a proof of Fermat's Last Theorem.

The full modularity theorem was finally proved in 1999 by Breuil, Conrad, Diamond, and Taylor who, building on Wiles' work, incrementally chipped away at the remaining cases

until the full result was proved.

Several theorems in number theory similar to Fermat's last theorem follow from the modularity theorem. For example: no cube can be written as a sum of two coprime n-th

powers, n 3. (The case n = 3 was already known by Euler.)

29

References

Henri Darmon: A Proof of the Full Shimura-Taniyama-Weil Conjecture Is Announced, Notices of the American Mathematical Society, Vol. 46 (1999), No. 11. Contains a gentle introduction to the theorem and an outline of the proof.

Christophe Breuil, Brian Conrad, Fred Diamond, Richard Taylor: On the modularity of elliptic curves over Q: Wild 3-adic exercises, Journal of the American Mathematical Society 14 (2001), pp. 843939. Contains the proof of the modularity theorem.

Barry Mazur, Number theory as gadfly- American Mathematical Monthly, 98 (7), August-September 1991, pp. 593610, Disscusses the Taniyama-Shimura conjecture 3 years before it was proven for infinitely many cases.

Weil, Andr ber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 168 1967 149-156.

Wiles, Andrew Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2) 141 (1995), no. 3, 443--551.

Taylor, Richard; Wiles, Andrew Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 (1995), no. 3, 553--572

Wiles, Andrew Modular forms, elliptic curves, and Fermat's last theorem. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zrich, 1994), 243--245, Birkhuser, Basel, 1995.

Retrieved from "http://en.wikipedia.org/wiki/Modularity_theorem"

30

P A R T 3

Possible variants proof of Fermats Last Theorem over Q

and applications Abel group, as alternative Wiles proof

Alternative statement

I offer to you attention a solution Fermats problem as solution of system of equations P. Fermat.

The are enough reasons to assume that P. Fermat

considered the equation:

nnn cba =+ (41)

31

as an infinite system of independent equations with finite number of variable n a t u r a l numbers (a, b, c, n) :

..................

..................cba

.................

.................cbacbacba

nnn

444

333

222

=+

=+=+=+

(42)

He knew that the equation (1) at n = 2 came from extreme antiquity. It has geometrical interpretation

and is called the equation of Pythagoras (approx. 580 500 B.C.):

222 cba =+ (43)

It was also known that among the infinity aggregate of solutions of equation (3) there are such threes

of numbers )c,b,a( 000 that do not have common multipliers. These threes of

n a t u r a l numbers are known as p r i m i t i v e t h r e e s of Pythagorean. These primitive threes of Pythagoras

were also known to be generated by any couple v > u of natural numbers

of different parity with the help of three invariant forms:

22

0

0

220

uvcuv2b

uva

+==

= (44)

32

The further line of argument is obvious.

Four variable natural numbers )n,c,b,a( 00o are building three forms:

n n2

0 Daa = (45) n n

20 Dbb = (46)

n n2

0 Dcc = (47) Here

3/)cba(D 2n02n

02n

0n ++= (48)

universal common multiplier

Key condition for primitive Pythagorean triplet: Diophantus and then Fibonacci , indicated the following method

of search of solutions of Pythagorean equation :

222 zyx =+ (49)

If whole numbers v and u are such that v > u and

greatest common divisor GCD (v, u)=1 , at that v and u of different evenness than triads ooo c,b,a , given by equations:

22

0

0

220

uvc

uv2buva

+==

= (50)

are primitive solutions of Pythagorean equation :

2o

2o

2o cba =+ (51)

N O T E Pair natural numbers

uv >

I s b a s I s f o r P r o b l e m s o f F e r m a t , R i e m a n n , B i r c hs a n d S w i n n e r t o ns D y e rs

See Diagram 1 and Plan 1

33

Diagram 1

All prime numbers settle down in a natural line in pairs,

intervals between which submit to a rhythm of numbers v=2 and 3ao = :

2

2

2131721113

2711257235123112

=======

2

2

3

2

243472414323741

323137223312192321719

===

====

27173263712616325761253572515324751

2

2

2

2

=======

2

2

2

2

2

2971012939728993287892838727983

327379

======

=

......................2117121211311721111132109111210710921031072101103

2

2

2

=======

(52)

Et cetera, et cetera

34

G e n e r a l c o n c l u s i o n

See formerly :

Basis for Inductive way is Abelian group of three whole numbers:

cCvectororccsquare

bBvectororbbsquare

aAvectororaasquare

2o

2o

2o

General basis for all rightly solutions to problems of Last theorem, Riemanns conjecture,

Birch and Swinnerton-Dyer conjecture is Abel Group f,G .

C o n s e q u e n c e 1

If v > u are the numbers of various evenness taken from endless series of natural numbers , vide supra, then :

n

o2n

o2

o2n

o2

on

2no

2o

no

2no

2o

n

2no

2o

2no

2o

no

n

cbcacC

cbbabB

cabaaA

++=++=++=

(53)

Three vectors over field Q natural numbers.

It is objects to Abel group f,G with binary dealership: Action associative

oooooo

oooooo

ooooo0

b)ac(baca)cb(acbc)ba(cba

===

(54)

Left -distributive

ooooooo caba)cb(a +=+ (55)

35

Right- distributive

ooooooo acaba)cb( +=+ (56)

The combination of this form the cornerstone of the motion of linearity, which underlies such fundamental tools as scalars and vectors

n

o2n

o2

o2n

o2

on

2no

2o

no

2no

2o

n

2no

2o

2no

2o

no

n

cbcacC

cbbabB

cabaaA

++=++=++=

(57)

with all this going on:

n

on1

o

no

n1o

no

n1o

c)c(

b)b(

a)a(

===

(58)

1Ecba 0o0

o0

o ==== (59)

mn

om

on

o

mno

mo

no

mno

mo

no

ccc

bbb

aaa

+

+

+

===

(60)

Pay attention.

Everywhere

n

mno

mno

mno D3)cba( =++ (61)

where

3/)cba(D mnomn

omn

on ++= (62)

36

Consequence 2

Linear combination

n n

o2n

o2

o2n

o2

on n

n 2no

2o

no

2n2o

n n

n 2no

2o

2no

2o

no

n n

3/)cbcac(CCc

3/)cbbab(BBb

3/)cabaa(AAa

++===++===++===

(63)

endless series roots to Fermats equations :

nnn CBA =+ (64)

P A R T 4

Non-modular elliptic curves,

10-th problem of D.Hilbert, and Rule A , Rule B , Rule C

as way to proof of Riemann Hypothesis

I propose:

1. Co-prime bases for zyx CBA =+ as bases for irrational roots by Fermats equation:

nnn cba =+ (65)

2. Instead of parity p by Frey:

)p(mod)bX()pX(XY qq2 + (66)

deterministic expression

)BX(X)AX(Y2 += (67)

In this case we have substitutions:

37

2

0

2o

20

c)BX(

bX

a)AX(

=+=

=

(68)

where

2o

2o

2o

aB

abA

==

(69)

and

)uv(c

)vu2(b)uv(a

220

0

220

+==

= (70)

primitive Pythagorean triplets for all

uv > (71)

are the numbers of various evenness , taken from endless series of natural numbers.


)ab(A 202

01 = if oo ab > (72)

will not divide into

22

222

0

0

)()2(

)122b(4)vu2b(416

=======

(73)

if number

2n4

ooo )cba(16 = (74)

is function a discriminant my elliptic curves , see (15) and (34):

8n2

ooo

2)cba( = (75)

At the same time )ab(A 20

201 = will not divide

3

22o

220

3)12a(9

)uva(927

====

=== (76)

38


+= 2

o

3o

ba427 (77)

where )b27a4( 2o

3o += (78)

discriminant for canonical form any elliptic curves: oo

32 bxaxY ++= (79) Endless series prime numbers:

002

02

01 abif)ab(A >= (80) is endless series of conductors-criterions

for separation non- modular elliptic curves.

As result I receive my equation for endless series to non-modular elliptic curves for 2n = :

2

o2

o2

o2 cbaY = (81)

and for 2n>

no

no

no

n cbaY = (82) In this case:

n

0

no

n0

c)BX(

bX

a)AX(

=+=

=

(83)

where

no

no

no

aB

abA

==

(84)

and

)uv(c

)vu2(b)uv(a

220

0

220

+==

= (85)

primitive Pythagorean triplets for all

uv > (86)

are the numbers of various evenness , taken from endless series of natural numbers.

Here every number from endless series numbers

)ab(A n0n

0 = (87) will not divide into

39

22

222

0

0

)()2(

)122b(4)vu2b(416

=======

(88)

and will not divide

333927 == (89)

In the end, if

5c4b3a

o

o

o

===

(90)

then we have a minimal discriminant for 2n = :

62550

2)543(2

)cba(

8

4

8

n2ooo

==

== (91)

where

0 (92)

As explanatory, for 15n = :

53

8

30

8

n2ooo

102739210.22

)543(2

)cba(

==

== (93)

Consequently, my curve is non-singular, my curve is elliptical curve.

2. Instead of Freys formula:

pp v1pa += (94)

I propose three of Rule for computation all simple (prime) numbers :

Rule A, Rule B and Rule C.

It is original and novel results analysis of knowledge data about natural and primes-numbers, about primitive triplets of Pythagorean, about equations of Pythagorean,

40

P.Fermats and G.Freys equations. General result three Rules: Rule A, Rule B and Rule for separate and calculate endless series prime-numbers. It is adequate

interpretation conclusion of Matiyasevich, who has proved 10-th problem of D.Hilbert, proves to be true:

All prime numbers are simple search (recalculation) of all of some natural numbers.

Three forms of Rules, in according to conception by D.Hilbert, [12] and [11], creates expansion endless series forms, which a useful to become of fractals, see forms (21)(24),

end primes-numbers. The forms contains spectral invariant 2 , invariant )5(R5

= ,

invariant )7(

R7

= and general invariant complex function

20

20

0000

ba

)bia()bia(ssS

+==+==

(95)

as basis for calculations of gtneral zeta function:

0)S1()S1(2Ssin2)S( 1SS = (96)

and my zeta function:

==n

SQsin2)1()S( s (97-8)

Here Q variable quantity of angle, see Pic.1 and Pic.2 : Q0 (98-9)

and ...,5,4,3,2,1n=

Remark

G.F.B. Riemann (1826 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function

...4/43/12/11)s( 555 +++++= called the Riemann Zeta function - see: http://www.claymath.org/millennium.)

Primitive Pythagorean triads ,which halt a value primes numbers by a rule A

R U L E A

A ) If 50 Ac

= primesnumbers , then according :

41

05

00004

00003

002

02

02

002

02

01


baif)ba(A

abif)ab(A

=>=>=>=>=

(99)

For 00 ab > , numbers

)ab(A 003 = (100)

are primesnumbers and

)ab(A 202

01 = (101)

are primesnumbers or

)ab( 202

0 (102)

will divide by controller

)7(R7

= (103)

For 00 ba > , numbers

)ba(A 004 = (104)

are primesnumbers and

)ba(A 202

02 = (105)

are primesnumbers or

)ab( 202

0 (106)

will divide by controller

)7(R7

= (107)

42

Primitive Pythagorean triads, which halt a value primes numbers by a rule B

R U L E B

If controller

)5(

R5

= (108) divide numbers 0c , then according :

05

00004

00003

002

02

02

002

02

01

cAbaif)ba(A

abif)ab(Abaif)ba(A

abif)ab(A

=>=>=>=>=

(109)

For 00 ab > , numbers

)ab(A 003 = (110) )ab(A 20

201 = (111)

are primsnumbers

For 00 ba > , numbers )ba(A 004 = (112) )ba(A 20

202 = (113)

are primsnumbers

Primitive Pythagorean triads ,which halt a value primes numbers by a rule C

R U L E

If

2

2

)7(R7

=

(114)

divide numbers

)ab( 202

0 (115)

43

and

)7(

R7

= (116) divide numbers

)ab( 00 (117) then are

05 cA = (118) primesnumbers

Explanatory or demonstration examples for calculations of prime-numbers useful formulas:

05

00004

00003

002

02

02

002

02

01


baif)ba(A

abif)ab(A

=>=>=>=>=

(119)

Lets take from the book [1], see page17, ready triads of primitive Pythagorean triads

:)c,a,b( 000

)61,11,60(:3)29,21,20(:2

)5,3,4(:1

)65,33,56(:6)37,35,12(:5

)13,5,12(:4

)65,63,16(:9)41,9,40(:8)17,15,8(:7

)73,55,48(:12)53,45,28(:11

)25,7,24(:10 (120-33)

Substitute these numbers into my formulas and you will get

a full set of prime-numbers, born by corresponding of following rules:

Primitive Pythagorean triads, which

halt a value primes numbers by a rule A

Triad 2 :

41400441)ba( 202

0 == prime number 12021)ba( 00 == prime number

29cA 05 == prime number

44

Triad 4 :

177/11925144)ab( 202

0 === prime number 134)ab( 00 == prime number


Triad 5 :

10811441225)ba( 202

0 == prime number 231235)ba( 00 == prime number


Triad 7 :

237/16164225)ba( 202

0 === prime number 7815)ba( 00 == prime number


Triad 8 :

2177/1519811600)ab( 202



Triad 12 :

1037/72123043025)ba( 202

0 === prime number 74855)ba( 00 == prime number


Primitive Pythagorean triads, which halt a value primes numbers by a rule B

45

Triad 6 :

135/65)5/( 0 == prime number 204710893136)ab( 20

20 == prime number

233356)ab( 00 == prime number

Triad 9 :

135/655/c0 == prime number 37132563969)ba( 20

20 == prime number

471663)ba( 00 == prime number

Triad 10 :

55/255/c0 == prime number 52749576)ab( 20

20 == prime number

17724)ab( 00 == prime number

Primitive Pythagorean triplets, which halt a primes numbers by a Rule

Triads 3 :

If 2

2

)7(R7

=

divide number

7149/34791213600)ab( 202

0 === prime number and

)7(R7

=

divide number

46

77/491160)ab( 00 === prime number then

61cA 05 == Prime number

The remainder example

Triad 13 : If v=17 and u=14 , then according (13):

485c476b93a

0

0

0

===

9272176898576226)ab( 202

0 == prime number 383)93476()ab( 00 == prime number

975/c0 = prime number It is Rule B

Triad 14 :

If v=17 and u=12 , then according (13):

433c408b145a

0

0

0

===

777207/43914502521464166)ab( 202


433c5A 0 == prime number It is Rule A

Triad 15 :


445c396b203a

0

0

0

===

60711520941816156)ab( 202

0 == prime number 193203396)ab( 00 == prime number

895/cA 05 == prim number It is Rule B

Triad 16 :


493c468b155a

0

0

0

===

857277/99919402524024219)ab( 202

0 === prime number

47

313155468)ab( 00 == prime number 493cA 05 == prime number

It is Rule A

Et cetera, et cetera .

P A R T 5

Intercommunication

a between the elliptical curves, Abel group

and the non-modular forms.

According [1], chapter 11, section A , the key moment is to connect information of elliptical curve of Frey with analytical function of complex variable and global invariant infinite

product of simple numbers p :

As realization that statement I propose a solution possible variant of equation Freys-Yaroshs:

)cba(

)BX(X)AX(Yn

on

on

o

2

==+=

where

n

0

no

n0

c)BX(

bX

a)AX(

=+=

=

And

22o

o

22o

uvcvu2b

uva

+==

=

primitive Pythagorean triplets.

In this case we have endless series non-modular curves.

Because:


)ab(A 202

01 = if oo ab > will not divide into

222

22

0

0

)()2(

)122b(4)vu2b(416

=======

48

If number

2

n4ooo )cba(16

=

is function a discriminant my elliptic curves:

8

n2ooo

2)cba( =

at the same time )ab(A 202

01 = will not divide

3

22o

220

3)12a(9

)uva(927

====

===


+= 2

o

3o

ba427

where )b27a4( 2o

3o +=

discriminant for canonical form any elliptic curves:

oo32 bxaxY ++=

With all this going on to make good use multitude

construction to analytical function of complex variable:

20

20

20

0000

ba

)bia()bia(ssS

=+==+==

and

nn0

n0

n0

n0

n0

n0nnn

zba

)bia()bia(ssS

=+==+==

as general common invariants for systems co-ordinates A and B It is complex functions.

I propose three of Rule for computation

all simple (prime) numbers : Rule A, Rule B and Rule C.

See Part 2

And following research result a Riemanns statements.

Riemanns statement 1

Function:

49

1)s1()s1(2sSin2)s( 1ss =

determines all complex numbers 1s

My statement 1

Function:

1)s1()s1(2sSin2)s( 1ss =

determines all complex numbers 1s if

1ba

)bia()bia(ssS2

02

02

0

0000

=+==+==

See Pic.2

where:

2o

2o

2o

22o

o

22o

cba

uvcvu2b

uva

=++=

==

For all pairs number uv > of various evenness taken from endless series of natural

numbers N= 1, 2, 3, 4, 5,..


)s(Z function determines the seros for ...,6,4,2s =

My statement 2

0SQSin)S(Z == function determines endless series zeros 0bo = for

...,S6,S4,S2S =

If angle Q=0 , see Pic.2 and Pic. A , then:

50

0bi0bi

o

o

==+

1a

)bia()bia(ssS2

0

0000

===+==

what

1cba

1uvc0vu2b

1uva

2o

2o

2o

22o

o

22o

==+=+=

====

where v=1 and u=0 taken from:

==

,...5,4,3,2,1,0u...,5,4,3,2,1v

Geometrical interpretation, see Pic. A

Pic. A


Forma:

=

= 1n sn)n(

)s(1

determines stripe

1)s(R0

51

for all untrivial zeros, when

Rt,ti21 +

My statement 3

Let 2S2 = where:

++++++= nif2...2...2222 n32102 then:

+++++=+= nif1....2....222.1 n321222 where:

r21

2 += and

++++= nif5.0....2...22r n32 Forma:

=

= 1n sn)n(

)s(1

determines stripe 1)s(R0

for all untrivial zeros, when

Rr,ri21 +

as equivalent:

R,i21 +

See Pic. B

Pic. B

52

Here:

==

==

==+

+=+=

+=+=

+=+=

............

5.1i21)i(z3intPo

1i21)i(z2intPo

5.0i21)i(z1intPo

...........

5.1i21)i(z3intPo

1i21)i(z2intPo

5.0i21)i(z1intPo

R e s u l t

Forma:

=

= 1n sn)n(

)s(1

determines stripe

1)z(R01)z(R0

for all untrivial zeros, see Pic.2 and Pic. A, when, angle 0aQ o == : 0)S1()S1(

2SSin2)S( 1SS ==

and complex function :

0b)bia()bia(ssS 2o0000 ==+== because:

0vu2bo == where v=1 and u=0 taken from:

==

,...5,4,3,2,1,0u...,5,4,3,2,1v

53

Consequently, exist endless series complex numbers

as endless series of zeros:

0)bia(iz0)bia(iz

oo

oo

===++=

Forma:

0n

)n()S(

11n

S=

=

contain function: +++++== nif1...2...222)n( n3212

and complex function:

1a

)bia()bia(ssS2

0

0000

===+==

what

0vu2b1uva

o

22o

====

where v=1 and u=0 taken from:

==

,...5,4,3,2,1,0u...,5,4,3,2,1v

Consequently, determines zeta function:

==

1n

S0

n)n(

1)S(

54

P A R T 6

Riemanns sphere as mapping of space task common solutions of Conjecture

Birch and Swinnerton-Dyer and Riemanns Hypothesis

1. Riemanns sphere

Lets address to Pic.3 and Pic.4.

If complex number oo bias += , see Pic. 2, is set on a plane ),( by a point )b,a(P oo the point )Y,,('P of crossing of a piece PN

of a straight line with a surface of Riemanns sphere is a new geometrical representation of number oo bias += in the Decartes system

of coordinates )Y,( . All complex numbers oo bias += and, see Pic.2, Pic.4, Pic.5, are displayed by points which projections

to a plane of complex numbers += iz and = i'z oo bias = lay inside of a semicircle of diameter, see Pic.3a.

Here

1)zRe(0

Pic.3a

Remark

Authors of Birch and Swinnerton-Dyer conjecture approve:

if (1) is equal to 0, then there are an infinite number of rational points (solutions), and conversely, if (1) is not equal to 0,

then there is only a finite number of such points.

55

2. General mathematical strategics a solution of the Birch and Swinnrton-Dyer Conjecture

and Riemanns Hypothesis

Defining role in the proof by Birch and Swinnerton-Dyer belongs to the EXPANDED or CLOSED plane of complex numbers:

+= iz and = iz properties of this plane are defined by properties of Riemanns Sphere

below, on Pic.4 Riemanns sphere [13] is represented.

Pic. 4

Here : Flatness ),( for complex numbers += iz and = i'z ;

Point )z,z(PP = ; Point )Y,,('P'P = ; Axis absciss for ; Axis ordinat for ;

It is stereographycal projection flatness ),( to sphere )Y,,(

Stereographycal projection keep angles . Thats have mathematical description, see (32):

2o

2o

2o

2o

2o

2o

2o

o2

o2

o

o

20

02

o2

0

o

c1c

ba1bay

c1b

ba1b

c1a

ba1a

+=+++=

+=++=+=++=

56

In a Fig. 5 it is presented diametrical section of Riemans sphere

Pic.5

3. INITIAL DATA


If angle =Q , ( see Pic.2 , Pic.3, and Pic.6) , then:

0)s(Re0)s(Re

ibib

o

o

==++

(9)


2o

oooo

ibaibassS

==+==

(10)

as endless series primitive numbers of Pythagora over field N natural numbers )uv( > :

)uv(S 22o +== (11) In the end we have endless products,

as endless series seros:

00sinsss,sS === (12) According this statement, we create endless series

functions

57

for proof a Birch and Swinnerton-Dyer conjecture

0)S1()S1(2Ssin2)S( 1SS == (13)

Here scalars 0S = are arguments for theyre functions .

Pic. 6

The axis of absciss oa is geometrical axis symmetry for two a system coordinat.see Pic. 6.

Common the angle Q is general argument for general functions: 0)S1()S1(

2Ssin2)S( 1SS == (14)


If angle 0Q = , see Pic. 7, then:

o

o

o

o

a)s(Rea)s(Re

0ib0ib

==

==+

(15)


20

0000

a

)bia()bia(ssS

==+==

(16)

as endless series primitive numbers of Pythagora over field N natural numbers:

58

)uv(Sa 22o == (17) In the end we have endless vectorials products, as endless series sero:

00sinss]s,s[S === (18) According this statement, we create endless series functions

for proof a Birch and Swinnerton-Dyer conjecture:

0)S1()S1(2Ssin2)S( 1SS == (19)

Here scalars 0S = are arguments for theyre equations. 0bi+

oib Pic. 7

Birch and Swinnerton-Dyer Conjecture

Mathematicians have always been fascinated by the problem of describing all solutions in whole numbers x,y,z to algebraic equations like

x2 + y2 = z2

Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult. Indeed, in 1970 Yu. V. Matiyasevich showed that Hilbert's tenth problem is unsolvable, i.e., there is no general method for determining when such equations have a solution in whole numbers. But in special cases one can hope to say something. When the solutions are the points of an abelian variety, the Birch and Swinnerton-Dyer conjecture asserts that the size of the group of rational points is related to the behavior of an associated zeta function (s) near the point s=1. In particular this amazing conjecture asserts that if (1) is equal to 0, then there are an infinite number of rational points (solutions), and

59

conversely, if (1) is not equal to 0, then there is only a finite number of such points.

4. SPECIAL PROPERTIES OF COMPLEX NUMBERS

The member of Russian Academy of Sciences, famous mathematician G. Pontryagin, see. [10] and Pic.2 Pic.7,

learning properties of complex numbers :

=+=

=+=

i'ziz

orbias

bias

oo

oo

(21)

discovered their polisemy, which was seen by G.V.Leibnitz in his time , as unexplained wonder.

Complex numbers can be in the same time: ) complex numbers,

) points representing these numbers on complex plane, ) vectors, corresponding to these numbers.

The length of such vectors is determined by module:

22

2o

2o

'zz

or

bass

+==

+==

(21)

Because of that there can be formulas given above.

Thus we have a possibility to consider primitive Pithagoras numbers 22

o uvc += as a basis of two forms of general complex invariants

(two forms of complex functions)

2

o2

o2

o2

o2

o c])bi(a([)ba(S ==+= (22)

220

20

0000

ba

)bia()bia(ssS

=+==+==

And also - as a basis of spectral invariant

60

++++++= nif...)c...ccc( no3o2o10oo (23) At that whole numbers

22o uvSc +== make a basis

for infinite numbers of whole-numbered decisions of the system of equation:

zyx

nnn

n22n

CBA2nifcba

,...3,2,1,0nif)uv()uv(f

=+=+=+=>

(24)

5. New type solutions to equations for Birch and Swinnerton-Dyer Conjecture

+==>

++++==>+++==>++==>

+==>==>

nif)uv(c)uv(f

.......................................................................................vu4uv4uv6uvc)uv(f

vu3uv3uvc)uv(f

uuv2vc)uv(f

uvc)uv(f1c)uv(f

n22non

464644884o4

2424663o3

42242o2

2211

00

(19)

6. General proof

Birch and Swinnerton Dyer Conjecture

If angle =Q , see Pic.2 and Pic.3, then:

0)s(Re0)s(Re

ibib

o

o

==++

(20)

and also to data of 3 7, on the EXPANDED complex plane of numbers: += iz and = i'z (21)

all real coordinates 0= . Hence, in the Decartes system of coordinates )Y,,( is formed a line

of infinite set of ZERO: 0)zRe( = and 0)'zRe( = (22)

Similar picture we have in the system of coordinates represented on Pic. 2: 0)sRe( = and 0)sRe( = (23

thus, in a point 1= , see Pic.5, function is formed:

61

0)1()1(2

sin2)( 1 == (24) on the basis of spectral invariant:

+=

=++++++==++++++=

nif1...)c...cc(1

...)c...ccc(n

o3

o2

o1

o

no

3o

2o

10

oo

(25)

From which definitions of ordinate follow:

)(1 == (26) and corresponding functions:

0)11()11(2

1sin2)1( 111 == (27) As a result we receive return display from the EXPANDED complex

plane ),( on a plane of complex numbers:

oo

oo

biasbias

=+=

(28)

systems of coordinates, represented on Pic.2. It means, that spectral :

+=

=++++++==++++++=

nif1...)c...cc(1

...)c...ccc(n

o3

o2

o1

o

no

3o

2o

10

oo

(29)

Is put inconformity ton infinite number of primitive Pythagoras numbers :

++++++

nif...)c...ccc( no

3o

2o

10

oo

(30)

and corresponding infinite row of integer decisions of the equations of the First type, see (19).

Remark.

Spectral invariant is formed over field N of natural numbers. It does not depend on a choice of coordinates and on operation

of display of complex numbers on the EXPANDED plane of complex numbers.

If

oo

oo

biasbias

=+=

(31)

then we have stereographycal projection

flatness ),( to sphere )Y,,( and keep angles, see Pic.4 .

62

Thats have mathematical description

1c1

cc1y

c11

c1c

c1

2o

2o

2o

2o

2o

2o

2o

+=+=

==

=

(32)

if

triads of Pythagorean:

22

0

0

220

uvcvu2b

uva

+==

= (33)

for all paar v>u numbers are the numbers of various evenness taken from endless series :

,...4,3,2,1)1u(v,....,4,3,2,1,0u

=+==

(34)

and , see (4) :

2

oo2

o

2o

o2o

2

ucu2

buav

vcv2

bavu

==+=

=== (35)

In the end, also:

If 2

ocn= then we create relativity system coordinat: For operation of display of my complex numbers, see (28), (31) and Pic.2,

on the EXPANDED plane of complex numbers:

==+=+=i'zbia'sizbias

oo

oo (37)

63

I offer two easy Rule:

==

+=+=

in

1nin1'z

in

1nin1z

n

n

(38)

for one in two endless series numbers:

;...n1ni

n1z;...,

32i

31z;

21i

21z;1z n321

+=+=+== (39)

;...n1ni

n1'z;...,

32i

31'z;

21i

21'z;1'z n321

==== (40) Endless series numbers (39) is enless series points:

n1

n = and n1n

n= (41)

Theyre lie to segment, see Pic.9: 1=+ (42)

Also if : n (43)

then: 0lim n = and 0lim n = (44)

and ii10)i(limzlim nnn =+=+= (45)

Concequence:

modulus 0izn = (46)

All theyre have geometrical interpretation,see Pic.8 :

Here 1)zRe(0 for +=+= ii)z(Rez

Pic.8

64

Here striple 1)zRe(0

Pic.9

C o r o l l a r y

B.Riemann (1859) :

(s) function determines all complex numbers 1S . Correspondence:

4. Special properties of complex numbers

(s) function determines the seros for ,,,,6,4,2S = Functional equation

(47)

and forma

(48) determines stripe

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