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Теория Чисел.
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.
-2001
-
. .
2 512.54
E78
22.1
..
78 . : . / ..,
2001. 37 c.
.
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- ;
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.
41.
1.1.
: 5- -
10 ?
:
1 2 3 4 5
10 .
, ,
10 . . 105 .
( :
0 1) 25 . k
n :
A = kn (1)
n (); k ().
: k -
( ) n
(, , ). , -
,
? (1).
. C 10-
? -
(0, 1 2),
. . 310 .
1. 5 , -
6 .
5 ( ),
, ?
2. .
, ,
?
3. 5 .
3 . -
, -
?
4. (-
) 1 0 30 .
1 0.
?
, -
: 210 > 103.
, . , ,
n i- ki -
. -
?
:
A = k1 k
2 k
3 k
n =
1
n
ii
k=
(2) . 100+200+400+800 -
3 ,
5, 6, (
). -
?
(2) , -
: 3 5 6 1 = 90.
1. ,
5 , 20 10
?
2. 7
0, 8, 9? (73 104)
63. , -
.
? (86400)
1.2.
: 5- -
, -
?
:
1 2 3 4 5
10 (0, 1, 2, 3, 4, 5, 6, 7,
8, 9). , ,
9 , 8
. . 10 9 8 7 6 = 30240 -
, .
, k n , -
,
:
A = n(n 1)(n 2)...(n k+1) = n!/(n k)! ( 3 )
(3) s! s, . .
1 s. , s! = 1 2 3 s.
. 25 ,
. -
? 25
. -
, 24 -
. 23 . -
: 25 24 23 = 25!/22! = 13800.
1. 25 -
, , . -
? (25 24 23 22 = 25!/(25 4)! = 303600)
2. -
(
7 ). -
, , -
, ,
12? (12!/9! = 1320)
3. 12 15 . -
. (
). ?
(15!/(15 12)!)
1.3.
, .
12, c .
, , -
n .
(3) n = k,
An
n = P
n = n! (4)
. 6 . -
?
6
. -
(4) 6! = 720.
1. ( ) -
? (8! = 40320)
2. 12 . -
, , ,
, . ,
.
? (12!)
3. 14 .
,
? (14!)
, , -
, -
3 7.
81.4.
,
. -
, .
n1 1- , n
2 2-, nk -
k- n1+ n
2 +...+ nk = n.
P(n1, n
2, n
3, ...n
k) = (n!)/(n
1! n
2!...n
k!) (5)
( 5 ) n
, .
n! , -
n1 , -
n2 -
. . (5).
. -
. 2 , 2 , 1 , 1
, 2 , 1 , 2 , 1 1 .
, : P(2,
2, 1, 1, 2, 1, 2, 1, 1) = 13!/(2! 2! 2! 2!)= 13!/16.
1. 2 , 4 1 . -
. ?
2. 5 , 4 2 , -
. -
?
3. . 5
, 6 3 -
. ,
-
?
4.
: y0, y
1, y
2, y
3. -
, 2 -
y0, 4 y
1, 3 y
2 1 y
3?
91.5.
. , -
. -
( -
) .
. 28 2 .
?
: (7 -
) (21 ). 6 -
, 12.
: 7 6 + 21 12 = 294.
2 28 378; . . -
7 9 (294/378 = 7/9) 2
.
, 4
4 , 2 .
, 33
: , , , , , , , , , , , , , , c, ?
,
, , 10
*?
1.6. ( )
. 67 . 48
, 35 27 .
, ?
. -
, ,
(67) A B 48 35
( ). -
27 -
, . -
, A, B.
10
67
A=48B=35
AB=27
, N = 67 48 35 + 27 = 11.
( -
)
N .
n . -
. -
: 1,
2,
3, ..., n. N(i) -
i - , ( , )i jN i , j. , - :
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( )( ) ( )
1 2 3 1 2 3
1 2 1 3 1 4
1 2 3 1 1 2 3
1 2 3
, , ,..., ...
...
... ...
1 ,..., .
n
n
n n n
ni
N N N N N
N N N N
N N N N
N
=
+ + + +
+ + + + +
+ ( 6 )
. 20 -
, 12 , 11 5
.
,
( -
), N = 67 48 35 20 + 27 + 12 + 11 5 = 9.
1 N. k? -
, [N/k], [x] x. , -
, k1, k
2, k
3...
. 1 100 5 7?
.
5,
11
7. 5 20 . 7 14 . 35 2 .
, 5 7: 100 20 14 + 2 = 68 .
1. 12 , 6 -
, 4 , 7
, 3 -
2 .
,
. ?
2. 3 , 2 , 1 4 -
.
, -
-
?
3. , 3, 5, 7,
200 500?
1.7.
k n, -
,
!!( )!
kn
nCk n k
=
(7)
(7) .
k n. n!/(n k)!
k
. -
. -
. k!
. 25 -
. , , -
, , 25 24 23 .
, -
, 3!
2 . 60- -
, :
5/36 6/49. , , 6/49. -
12
, 49 . -
- . () 6
. -
. -
, , -
. , (
), .
, -
, 6/49. -
, , :
49 48 47 46 45 44. ,
:
49! 139838166!43!
knC = = .
. -
1, -
0. -
. 6
() 49 6 = 43 (), . .
( ) 49!6,43 13983816.6!43!
P = =
-,
14 6 . -
5 ?
( 16C = 6)
( 143C = 43). : 6 43 = 258 14 5 . 4 ? 6 -
43
. :
2 66 43
6 5 43 42 13545 .1 2 1 2
C C = =
, 3 246820 , . . -
1,77% . , 60
,
13
, 4, 5, 6 .
, ,
, .
-
. ,
- , , -
,
-
- (,
). 6, 5, 4 3
. , , -
: ,
. . -
, , -
, , . -
30 .
, -
-
, .
-
. ,
, -
3 -
. ,
. .
3 2 4.
, , -
. ,
. ,
- ,
, , !
-
.
,
, -
3 . , -
4 ?
14
1.8.
. 7 .
: , , .
? :
, ().
. , -
: 1101110101. -
.
,
, -
,
. 2 , 3 -
, 1 1 .
7 () 3 -
().
n (
) ,
, k -
? 0 1, 1 -
, 0 .
k n 1 , ,
k
. k n 1
, , -
k .
P(k, n 1) = (k + n 1)!/(k!(n 1)!) = 1kn kC + . (8)
1. 5
, , a, b, c, d, f. 11 -
. -
, , -
, ? , baabcf
fabcba . ,
(8) ,
.
15
2. -
.
. - -
, 6 , -
, ?
1.9.
, -
.
1. .k n kn nC C
=
2. 11 1.k k kn n nC C C
= +
3. 0 1 2 ... 2 .n nn n n nC C C C+ + + = . ,
(7). -
. -
, n = 2 :0 1 2 22 2 2 1 2 1 4 2 .C C C+ + = + + + =
n = 3 :0 1 2 3 33 3 3 3 1 3 3 1 2 .C C C C+ + + = + + + =
1. n 2n . -
5 , ,
2 4 ?
2.
. ,
32 ?
1.10.
, -
.
a) . 5 4 .
, ,
. ( -
6) . ,
16
, 46C = 15. n k : 1
knC +
) . m , ,
k , .
k , nk. (nk+1 -
). k . :
1kn kC + ( 9 )
) . 12 .
5 , , .
-
,
? : 15 +21 = 36. n ,
k, , -
n 2k.
), ) ) .
1.11. ( )
5 . -
, -
? -
: N( 5 ) = 5! 15C 4! + 25C 3!
35C 2! +
45C 1!
55C 0! = 44.
-
, ,
. 5!,
. -
,
. . 5 .
1.12.
, -
, . -
, -
, .
17
n :
N( n ) = N 1nC N(1) + 2nC N(2) ... + ( 1)n N(n)
.
n ,
:
N( n ) = n! 1nC (n 1 )! + 2nC (n 2)! ... + ( 1)n 0! = Dn (10)
Dn. -
. Dn :
( )11 1 1! 1 ...1! 2! 3! !
n
nD nn
= + + + ,
[...] e1 -
n.
, -
. ,
n! = (n1)[(n1)! + (n2)!] ,
Dn = (n1)[Dn1 +Dn2] .
Dn = nD
n1 + (1)n .
:
n 1 2 3 4 5 6 7 8 9
Dn
0 1 2 9 44 255 1784 14273 128456
1. n = 10, 11, 12.
2. 6 .
2 , ?
3. 10 , . -
, 4 -
( ).
? -
, 6
?
18
1.13.
, -
. 9
. ,
, .
: (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7),
(7, 8), (8, 9). .
, . -
. -
N = 9!.
. 8. -
, 8 :
N( 8 ) = 9! 18C 8! + 28C 7!
38C 6! + +
48C 1! = 142729 = D9+D8
n
Dn + D
n 1.
a1, a
2, , a
10. -
-
, . . (a1, a
2), (a
2, a
3), , (a
9, a
10)?
1.14.
-
. -
.
3 ( -
) 2 . 32 , . .
10 .
:
( )332!
.
10! 2!N =
32! . -
10 , ,
19
, 2 .
, 10
, 2 .
32! 10! 2!
n c
3 . ?
n
2 .
, -
. , -
n . -
, :
P(n, 2) = (n + 2)!/(n!2!).
,
(, ) ?
, n k -
, 1- (, ) n1
,
n2 , k- nk , n1+ n2+ ...+ nk = n,
( )1 21 2
!, ,..., .
! !... !k knP n n n
n n n=
1.15. N
-
, . -
5 ,
. 5
? .
1- 5 ., 4 ., 3 ., 2 ., 1 ., 0 .
2- 0 ., 1 ., 2 ., 3 ., 4 ., 5 .
6 . n
2 , n + 1 .
2 (-
, , ),
.
20
. : 10 , 15 ,
12 . 2 . 2
11 , 16, 13 -
. ,
: 11 16 13.
. n1 1-
, n2 2-, ... nk k-.
2 .
(n1+1)(n
2+1)...(nk+1).
N. -
N.
. N , . . 1 21 1 ,...,knn n
kN p p p= . N -
2 : . . :
(n1+1)(n
2+1)...(nk+1).
. N = 600 = 233152. (3+1)(1+1)(2+1) = 24.
1. : 1350, 1617, 8280, 10013?
2. ?
3. 5 , 4 -
6 ?
4. , 2 -
5 , 10 . 4 -
, 3 , .
?
-
.
1. n k ,
s .
n k:
( )!
! !kn
nCk n k
=
s k sn sC
. . -
21
.
k k sn n sC C
2. 15 ,
5 . ?
, . . -
1, 2, 3, 4 . :
1 2 3 415 15 15 15 15 105 455 1365 1940.C C C C+ + + = + + + =
215
1. -
.
1. ,
n , k .
2.
) 5, 6, 7; )3, 0, 9; ) 5, 7 ?
3. ,
1, 2, 3, 4. ,
.
4. 7 .
, ,
4 ,
?
-
. n k -
n, k
( n k ). :
P(k, n k). , , 5 3:
1) 1 1 1 0 0
2) 1 1 0 1 0
3) 1 1 0 0 1
4) 1 0 1 1 0
5) 1 0 1 0 1
6) 1 0 0 1 1
7) 0 1 1 1 0
8) 0 1 1 0 1
9) 0 1 0 1 1
10) 0 0 1 1 1
22
, 6 4 .
, ,
00. ,
000 .
1), 2), , 10) ,
-
, 5
h = 3.
7 4, .
n h. , -
n k,
P(n, n k) = n!/ ((n k)!k!), ,
h = n k + 1.
, , n = 4, h = 3, . . 4,
, 3 -
. k = n h + 1.
k = 4 3 + 1 = 2; . .
4 2:
1) 1 1 0 0
2) 1 0 1 0
3) 1 0 0 1
4) 0 1 1 0
5) 0 1 0 1
6) 0 0 1 1
, 6 ,
, . . ( ) 1, 2 3
, 1, 4 5 , -
2, 4 6 3, 5 6
.
1.16.
. 40 .
?
40 2 , -
, -
23
.
40 3 . -
, 40 2 . -
:
P(40, 2) = 42!/(2!40!) = 861.
n1 1- , n
2 -
, , nk k- s ,
P(n1, s 1) P(n
2, s 1)P(n
k, s 1) .
,
- ,
.
(, ), -
.
1. , , 5 , -
8 . -
:
) , ,
) , (
5).
2. a) ) n
k .
3. : A a1, a
2, , an B -
b1, b
2, , bk.
A B
, s , ,
p A.
:
, 7 4 6
, 2 .
1.17. :
n k . -
, ?
24
, , -
.
, . :
P(n, k 1) = (n + k 1)!/(n!(k 1)!).
n! , - ,
n! ,
.
, -
, , ,
, , , s -
n.
( )1
! , 1 .n
is P s k
=
1.18. :
n
k . .
,
, , ,
. . ,
, n k. ,
:
(s s )
, -
.
( k + 1)
( , )
, ( ). -
, n k + 1 , -
. n k + 1 ( -
) .
,
.
25
, , -
P(n 1, k + 1), P(n, k);
. ., ,
,
P(n, k) P(n 1, k + 1).
, n = 4 k = 5
P(4, 5) P(3, 4) = 9!/(4!5!) 7!/(3!4!) = 126 35 = 91.
1.19.
1. .
4, 6, 10 .
, 18 . -
, , . .
4, 10, 4 10, 4, 4 .
:
F(N) = F(N 4) + F(N 6) + F(N 10),
F(N)
N. F(N) N.
F(N) = 0 N
26
0, -
. -
. :
F(1, 2, 3, 5, 10, 15, 20; 23) = F(1, 2, 3, 5, 10, 15; 3) + F(1, 2, 3, 5, 10, 15; 23)
5, 10 15 -
,
, . . :
F(1, 2, 3; 3) + F(1, 2, 3, 5, 10, 15; 23) =
=F(1, 2; 0) + F(1, 2;3 ) + F(1, 2, 3, 5, 10; 8) + F(1, 2, 3, 5, 10; 23) =
= 1+ F(1; 1) + F(1; 3) +F(1, 2, 3, 5; 8) + F(1, 2, 3, 5, 10; 23).
, F(1, 2; 0) = 1; F(1, 2; 3) = F(1;1) = 1; F(1; 3) = 0;
F(1, 2, 3, 5, 10; 23) = 0. :
1 + 1 + 0 + F(1, 2, 3, 5; 8) + 0 = 2 + F(1, 2, 3; 3) + F(1, 2, 3; 8) = 2 + 2 + 0 =
= 4. , 4 .
, .
3. (10 ). ,
, , -
: 1, 2, 3, 5 .
S(1, 2, 3, 5; 10 ) = S(1, 2, 3; 10) + S(1, 2, 3; 5) + S(1, 2, 3; 0).
-
, -
. 20 :
5 2 5+1 5 3+2 3+1 2 4+1 2
5+3+2 3 3+1 3+2 2+1 3 2 3+1 4
5+3+1 2 3 2+2 2 3+2+1 5 2 2+1 5
5+2 2+1 3 2+2+1 2 3+1 7 2+1 8
5+2+1 3 3 2+1 4 2 5 1 10
27
2.
2.1.
n [ 4 ]. -
:
h = 1 2
1 1
...
...
n
n n
a a a
a a a
,
2, 3, ... n .
hn = h0. (, , -
) n.
,
k ,
.
( )!knC n k -
.
2.2.
- -
. -
.
a) : .
,
, . -
. ,
28
-
. ,
. -
, 21 : 10
. -
: 21!(10!10!). ,
.
2 , ,
, ,
. -
: 2(21!/(10!10!) 20!/(10!10!) = 41 20!/(10!10!). -
(, 10 -
, ). :
22!/(10!10!2!). : P = 0,177. -
, .
) -
1000 , -
. , , -
100 , 4 .
, 1000 ,
.
,
: 4100C = 100!/(4!96!).
. , -
P = 24/94109400 = 0,000000255.
,
1000 :
P(1000) = 1 (1 0, 000000255)1000 = 0,0003.
) 8 ,
, . . . :
1. , .
2. , 8 3 -
, .
8 . 8! = 40320. 1- -
29
, , -
, ,
P = 1/40320 = 0,0000248.
P = 38C D5 = 56 44 = 2464.
3 8 P = 2464/40320 = 0,061.
2.3.
.
-
.
, , -
, . -
.
. -
, , -
, s .
(. . ). n .
n = 26, n = 33,
-
, . . , -
, . . 26 33 .
DES , 56 .
256 -
( ,
). -
, -
.
-
A = (a1, a
2, ..., an).
ai ,
( ). -
, n
k.
30
2.4.
. 5 -
, . -
, 8 ().
, 0,25 . -
, -
. , , ,
.
.
( 0,25 ), -
5 . 5 0,25, 20. -
20 7 . -
-
. :
111110111001001111111110011. -
:
1- 1,25 ,
2- 0,75 ,
3- 0,
4- 0,25 ,
5- 0,
6- 2,25 ,
7- 0,
8- 0,5 .
, -
-
, 20 7 .
,
P(20, 7) = 27!/(20!7!) = 888030.
, .
, -
.
2.5.
, , .
31
1948 . (K. Shannon) -
, -
-
, -
,
.
-
. x
1, x
2, , xn p1, p2, , pn.
( )1
log .n
k kk
H P P=
=
,
.
0 1 1 0 -
.
2.6.
. , (-
) (
) .
-
, -
. , -
-
, :
( )T
T T TX YL X,Y = ,
X Y+Y Y-X Y X, Y , XT, YT
(-).
,
-
.
, . .
n ,
, ,
,
- . -
-
n .
n--
, ( ). -
, 3
. , -
:
1 2 3n n nC C C+ + .
-
,
, n- . -
:
( )1 2 3 2 .nn n nC C C K+ + <
( )1 2 32 .n n n nK C C C< + +, n = 10 K
3. , -
: X: 10111001 Y: 00101111. , -
. X
, Y
,
. Y
X
Y.
2.7.
-
. -
,
: , , -
, , . -
: , , -
, , , .
, -
.
1. n . -
, -
? n () -
,
. ( ) -
( n 1), n,
n(n 1)/2.
2.
.
3. -
.
34
, ,
. ,
,
. , -
. -
, , -
, . -
,
-
.
,
.
-
.
[2, 3].
1. . . . .: , 1975.
208 .
2. . : . . .:
, 1963. 287 .
3. . : . . .: , 1970. 424 .
4. . . : . /
. ., 1998. 38 .
5. ., . : . .
.: , 1978. 412 .
35
................................................................................................... 3
1. ................................ 4
1.1. ....................................................... 4
1.2. ........................................................ 6
1.3. ..................................................... 7
1.4. .................................................... 8
1.5. ............................................. 9
1.6. (
) ................................................................................. 9
1.7. ........................................................... 11
1.8. .......................................................... 14
1.9. .......................................................... 15
1.10. ................................ 15
1.11. ( ) ...................................... 16
1.12. ....... 17
1.13. ....................................................................... 18
1.14. ......................................................... 18
1.15. N ................................................. 19
1.16. ............................... 23
1.17. : ............................................................. 24
1.18. : ............................................................ 24
1.19. ....................... 25
2.
.................................... 28
2.1. .................................................................................. 28
2.2. .................................................................... 28
2.3. ................................................................................ 30
2.4. ...................................................................................... 31
2.5. ....................................................................... 32
2.6. ................................................................ 32
2.7. ................................................................................ 34
............................................................................................... 35
..................................................................... 36
.
. .
. . , . .
020341 07.05.97. 04.12.00. 28.04.01
6084 1/16. . 3. . . . . 2,09. . .-. 2,22.. -. . 2,25. 100 .
-
-
190000, -, . . , 67
: . . . . 1998.
. . . . . 2001.
. . .
.
. . . .
. . . . -
. .
. . . .
. . .
.