akritas book russian copy
TRANSCRIPT
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(1) (2) (3) 1986 . . 1978 .
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(Sylvester, 1853). , (Vincent, 1836).
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(Pavelle et al., 1981).
12
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1.1, . 216 ; 1 232 ; 1, , , , 16 32 . . t f f= di ;1 L f 2 F f 6= 0, ( , , d1 + d2 + 2 F + dtt , | . L U], .e
| , ,
,
F
,
tL U
di i = 1 2 : : : t, e U . , ,
d1 6= 0
0 )
+ +
.. .. . 2( ; 1) t;1(U ; L + 1) + 1 ( ) . . ,
. ,
F ,
F
13
U = 2,
33-
F . 1.1.1.
, . = 2 t = 3 L = ;1
;4
;3
;2
;1 ; 1 0 1 1 1 2 4 2 ;1 4
2
3
4
. 1.1.1.
Computer methods for mathematical computations, 1977, p. 12. PrenticeHall, Inc., Englewood Cli s, New Jersey.)
;1 U = 2. (Forsythe, G.E., M.A. Malcolm, C.B. Moler.
=2 t=3 L=
f1 f2 )
F
, . F
( ( . F
F
) . , , -
, 335=4 + (3=8 + 3=8) = 2, 5/4, 3/8 2 (5=4 + 3=8) + 3=8 6= 2, F 7/4. , s, , , ( , ,
F. (5=4 + 3=8) 3/2, -
s
, . ),
.
14
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, , . 1.2) . , . . 1.3
, ). , ( (
-
-
1.2
:, , , ( . 2) ( .
) , -
.
,
(Horowitz et al., 1976). ai , x. ( (a1 a2 : : : an), n a = (a1 a2 : : : an), , ,
.0, 0 S, a x, ,
S
n = 0. a a
S : (1) a
(2) | -
,
15
: (a) = (a) = a1 (a) = an (a) = (a2 a3 : : : an) (a) = (an : : : a1) b1 : : : bk a k 1, (b1 : : : bk a1 : : : an) b1 : : : bk a k n, , bi ai , 1 i k, a = (ak+1 : : : an). a = 0, , a1 a a = (a1 ). n S. , , , ij , i= ij , , . i (i) = 1, in . , : , + .
.
.) a = (a1 a2 : : : an), ,
a
.. ..
, 1, i 0 in < 0,
, ,
j
P
0 j n ij
jij j
j
. i = (i0 i1 : : : in) n ; 1, , i>0 , , . , -
;1 = 2 ;1 |. in , , in | , .
, ..
16
?
,
, i = +23456789. . 1.2.1.
= 103 , . .
,
,
i ;! 789 ;! 456 ;! +23
. 1.2.1., . 1.2.1 . . , n=d n d, , )
i = +23456789 i. r = (n d), , , , .. , (
n d{
,
.
,
. ,|
. ( S, ), :
. 1.2.1 . 1.2.2. T E T, S . (x1 x2 : : : xn). n , (i + 1).) , , , E
. 1.2.2., 0 . inxi
xi , 0. ( ,
. (1 i < n), , ixi i-
17
xi
xi | . 1.2.1,
, ,
(x1 x2 : : : xn) |
ixi .
. i, . 1.2.3. 0
i ;! 0
789
S
;! 0
456
S
;! 0
+23
. 1.2.3.. 1.2.1. , . . 1 2 , , / . , , .. . 234, , . n = 123456789 , 1, 3 5 , 432, ( . , , . ,
) , -
.
,
1-234-56789, :
n = 123456789 105, r1 = 56789. q1 = 1234 103 , r2 ).
q1 = 1234 q2 = 1 r2 = 234 (
18
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np = 1432. , np = 1432 143256789
q2 = 1 1000 234), 105 , . . ( . f ( ,
432 ( r1 = 56789, .
-
| -
.,
,
). , ,
-
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g| S. , (1) f g, f = O(g), c1 x1 2 S, , jf(x)j c1 jg(x)j x 2 S x > x1 (2) f g, f = (g), g f, (3) f g , f g, f = O(g) f = (g). , , . 1.2.2. i L (i). dxe | , x, bxc | | x, 1 L (i) = dlog (jij + 1)e = blog jijc + 1 , i=0 i 6= 0: = 103 . L -
1.2.1.
.
i = +23456789 L (i) = 3, . . , , L L (
. 1.2.1 | ). L
19
1.2.3. An, | , S.
A| A., . ,
S| tA (n) n2S|
,
A
, ,
-
.3
m,
p(n) = O(nm )..
1.2.4.
.
p(n) = pm nm + +p1 n+p0 | n 1, p p0 + njm1j1 + jnmj nm ; c = jpm j + ( ), . ., , . 1.2.4 , -
jp(n)j jpm jnm + + jp1 jn + jp0j (jpm j + + jp1j + jp0 j)nm :+ jp1j + jp0 j. k O(nm ), O(n3) , , .. ci nmi 1 i k. m| tA (n) = O(n13)
jpm j +
mi 1 i k. n| ),
A
(..
tA (n) = -
A,
-
tA (n) = O(2n). ,
O(log n) < O(n) < O(n log n) < O(n2) < O(2n) , . , ,
20
?
, , , .. .
.
,
, | -
-
.. + .. , , . , . i1 ; ! ;;; i2 ; ! ;;; i2 . , , (integer summation), : , + , .. , , , / . i1 , , ,
, i2 ,
; ! ;;; ; ! ;;;
; ! ;;; ; ! ;;;
; ! ;;;i1
1. 2.
ISUM,+, . . , s. , , .
i1 i2 ,
+, , -
ISUM. (, .)
21
( +),
, ,
s = i1 + i2 ) , i1 ( i2 . ( , , 1.2.3).
-
.) Ofmax L(i1 ) L(i2 )]g max L(i1 ) L(i2 )] ( )( . , ),
tISUM (i1 i2) = , ,
, max L(i1 ) L(i2 )] + 1.
i1 i2 ( tIMULT (i1 i2) = O L(i1 ) L(i2)]. . i1 i2 q r, i1 = i2 q + r, 0 r < i2 .
(k + 1)m = (m0 m1 : : : mk ) kn = (n1 n2 : : : nk ) n m < bn, b| ( b= , ). b | 232 2 , m = m0 bk + + mk n = n1 bk;1 + + nk . , 1234 23, 123 ( m) 23 ( n), 5 8 84 ( m) 23, 3 15 . , . , q m n qt. n1 , n, m0 b+m1 qt . , b;1 qt = b m0 b+m1 c n1 n1 = m0 n1 > m0
22
?
qt :
.
b ; 1 qt n1 m0 b + m1 . ( n1 < m0 ?) q.
b = 10 , qt = 9. q = 6. , qt = 9. -
n = 69 m = 600, n1 = m0 , q = 8. n = 69 m = 480, qt = 48=6 = 8. n = 29 m = 200, n1 = m0 , q = 6.
n m < bn, n = 59 m = 600 ? (
n = 60 m = 600. . 1.2.5 .) , , qt , n = 69 , n = 29. , . 1.2.5. b| , m = m0 bk + +mk n = n1 bk;1 + +nk n m < bn. qt q ( | ) m n, qt q , n1 b=2, qt ; 2 q qt , qt q, q + 1, q + 2. . , m = m0 bk + m1 bk;1 + m2 bk;2 + + mk n = n1bk;1 + n2bk;2 + + nk n m < bn. qn m, m2 bk;2 + + mk < bk;1, k;1 < (m0 b + m1 + 1)bk;1, , qn1 b , qn1 m0 b + m1 , q b ; 1. , , qt b ; 1, , n1 > m0 , n1 , m0 b + m1 . , qt q. , n1 b=2 , (qt ; 2)n m. n2bk;2 + + nk < bk;1, (qt ; 2)n < (qt ; 2)(n1 + 1)bk;1 = qtn1 + (qt ; 2 ; 2n1)]bk;1 (m0 b + m1 )bk;1 + (qt ; 2 ; 2n1 )bk;1 qt. n1 b=2 qt b ; 1, qt ; 2 ; 2n1 < 0, (m0 b + m1 )bk;1 (m0 b+m1 )bk;1 +m2 bk;2 + +mk = m. , .
23
b=2, 2e { 2e m 2e n. e, . , 2, .
,
b = 10, n = 29 m = 200. , 2e 29 < 1000 e=5 n m 928 (= 32 29) 6400 (= 32 200) , 2e. 1.2.5 b . qt 1 , , n. 3242. qt = b272=32c = 8 qt = b134=32c = 4 qt = b50=32c = 1 qt = b176=32c = 5 qt = b139=32c = 4 , . . ..
2,
, , .. m n 2e n < bk+1.
2e ,
.
m, 272828282
-
272 828282 32 42
25936 84154 134 68 12968 50 02 3242 176 08 16210 139 82 12968 1014
. ,
, ,
(Knuth, 1981, p. 235{238, 246) 1984, p. 342{357)]. , ,
.
. 19{21
-
(Flanders, ,
24
?
i1
tIDIV (i1 i2) = O L(i2 )fL(i1 ) ; L(i2 ) + 1g] . . i2 (i1 i2 ), q r, , i2 q . . .
4 -
.
n r 1, p(x) n ( 0
p(x) p = (x cr er cr;1 er;1 : : : c1 e1), ci 6= 0, ci = (ci1 ci2 ::: cimi ), mi 1 ei er > er;1 > > e1 . n = er , , p(x) cr . p(x) p = (x n cn cn;1 : : : c0) . .] , ) . , p(x) = x3 ; 7 , . 1.2.4. p(x) = 0]
.
p; 1 S !
; 1 S ; 0 S ; 1 S ; 0 0 ! ! ! !0 0 +1
0
? ? yx
0
? ? y
+3
0
0 ;7 , .
? ? y
0
. 1.2.4.x3 ; 7 x x. | , , p(x) = 0.
25
x1 x2 : : : xv x1 x2 : : : xv;1.
, .. ci ,
v v;1 .
xv
..
1.2.6.,
p(x) =
P
0 i n cix
i
| ,
. (
P
0 i n jcij,
( ,
1)
.) p(x) | |
( jp(x)j1 = max0 i n(jci j), 1) | jp(x)j1 = jp(x)j2 = (P0 i n jci j2)1=2. , ,
, jp(x)j1 jp(x)j1 (n + 1)jp(x)j1, p(x) L jp(x)j1] L jp(x)j1]. P i p2(x) = P i p1 (x) = 0 i m ci x 0 i n di x | m n . m n L jp1(x)j1 ] L jp2(x)j1 ] , p1 (x) p2(x), p1(x) p2 (x) q(x) r(x), p1(x) = p2 (x)q(x) + r(x), r(x) < n ( , , m n). , , PSUM, (polynomial summation) p1(x) p2 (x) p1(x) + p2 (x) . , 2 max(m n) + 1] | , p1 (x)+p2(x).] , max(m n) + 1 . , . n|
26
?
Ofmax L(i1 ) L(i2 )]g.
i1 i2 p1(x)
tISUM (i1 i2) = ,
jp1(x)j1 p1(x)], p2(x) jp2(x)j1 p2(x)] , Ofmax Ljp1(x)j1 Ljp2(x)j1 ]g. max(m n) + 1 ,
tPSUM p1(x) p2(x)] = O( max(m n) + 1] maxfL jp1(x)j1] L jp2(x)j1 ]g): , , p1(x) p2 (x), tPMULT p1(x) p2(x)] = Of(m + 1)(n + 1)L jp1(x)j1 ]L jp2(x)j1 ]g: , p1 (x) | , p2(x) | , , .( .)
1.3, , , EUROSAM | .1) 1).
. , SYMSAC | ..
, -
-
EUROCAL
, , (Petricle (ed), 1971). , SIGSAM (Special Interest Group on 1988 . SIGSAM , ISSAC. | . .
27
Symbolic and Algebraic Manipulation) for Computing Machinary). Symbolic Computation. . 10 20,
ACM (Association
Journal of
1847 . .
10 . 1970 . . . , .. . , , -
-
,
.
,
,
, . : camal | . .
schoonship | sac-2 |1),
altran, sac-1
, -
sac-1 (symbolic and algebraic calculation{ Sac-1 |. . ( . , , (.. , |
, ) ,
): (1)
28
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(2) (4) , ,
(6) , ,
,
,
(5)
(3)
(7) (8) , -
sac-1,. , PSUM ,
.
sac-1
,
sac-1 ISUM |-
(polynomial summation), (integer summation). , .
-
. | macsyma, reduce, schratchpad, maple ( , reduce | : (1) (2) (3) (4) (5) (6) (7) reduce. , , maple. 1980- . 1960- . , . , -math. , , , , , , ,
.
). , , , -
29
, . 1987 .) HP-28c , , ( . Hewlett-Packard .
,
-
200
1.2 1. a. b.( n, ,.
n O(
En,n!, 1.2.1), tEn (k n).
, 2 3, 5 tF(n)? 1 -
F
4,
2.
.)
, n i3 = n(n + 1)=2]2: , ( ( ) ) , , ,
Xn O-
1 i n
-
a. b. 3.
,
. n
30
?
. n. ( , $(n).
, ,.
,
-
. 2.3.1. , ,,
n!1
lim $(n)=(n= ln n) = 1 n= ln n $(n).) n-
n
4.
. 1+2+3+
, + n = a n2 + b n + c n .
n -
a b c. 1, 2 3 12 + 22 + 32 + n -
5.. ( . + n2 = a n3 + b n2 + c n + d.)
,
1.2 1.. , , , , | -
31
;;;;;;; x ;
:
; ;;;;;;; ;;;; y ! ;
; ! ;;;;
( ), . (a) ( (c) . (d) , , , ( 300 , .. , , .. ) (b) ,
) ,
2.
. , i1 = (a0 : : : an;1) i2 = (b0 : : : bm;1) | , , L(i1 ) = n L(i2 ) = m, tIDIV (i1 i2 ) = O m(n ; m + 1)]. . , L(q) n ; m + 1, q = qn;m : : : q0 qi m .] 5. , , , , . . (x cr er cr;1 er;1 : : : c1 e1 ) r 1, ci , ei | . (a){(d) . 2 , ) (b) p1 (x) + p2 (x) ( , (c) (a) -
3. 4.
(d)
6.
p1 (x) p2 (x) . P p(x1 x2 : : : xv ) = 0 j n pj (x1 x2 : : : xv;1) (xv )j | v
32
?
#i p(x)]
v. v jp(x)j1 jp(x)j1 : v = 0, jp(x)j1 = jp(x)j1 = jp(x)j, jp(x)j | . v > 0 jp(x)j1 = max0 j n jpj (x)j1 ( ) jp(x)j1 = P0 j n jpj (x)j1 ( ). a. , L jp(x)j1] L jp(x)j1]. , p1(x) p2 (x) p1 (x) + p2(x) O maxfL jp1(x)j1 ] L jp2(x)j1 ]g Y (maxf#i p1(x)] #i p2 (x)]g + 1)]1 i v
p(x)
xi . p(x1 x2 : : : xv )
-
-
b.
c.
O L jp1(x)j1 ] L jp2(x)j1 ] Y f#i p1(x)] + 1gf#i p2(x)] + 1g :1 i v
;
p1 (x) p2 (x)
Flanders H. Scienti c Pascal. Reston, VA, 1984. Forsythe G.E., Malcolm M.A., Moler C.B. Computer methods for mathematical computations. Prence-Hall, Englewood Cli s, NJ, 1977. Horowitz E., Sahni S. Fundamentals of data structures. Computer Science Press, Rockville, MD, 1976. Knuth D. The art of computer programming. Vol. 2: Seminumerical algorithms. Addison-Wesley, Reading, MA, 1981. : . . . 2. { .: . 1977.] Pavelle R., Rothstein M., Fitch J. Computer algebra. Scienti c American. 136{152, December 1981.
33
Petricle S.R. (ed.) Proceedings of the 2nd symposium on symbolic and algebraic manipulation. Association for Computing Machinary, New York, NY, 1971.
II
, ,
,
, , | . , . III
. . , , | .
-
.2 .
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. -
2
(
)
|
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-
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,
2.1, , (Sims, 1984). . -
2.1.1., . x| | . .1 . x2A A x2A = B A B), A B A| x x x| B, A A A , A.-
.
-
A B(
,x2A , A = B,
x 2 B.
36
B
A. A| A B A 6= B ( , . .x2 = , P(x). , fxg, .
, ). x. , .. , A .
,A B B, , A. , x -
f x : P (x) g f: : : g.|Z= f : : : Q=
.
,
f 0 1 2 : :: g,0 g.Z, Q R
;2 ;1 0 1 2 : : : g, f x=y : x y 2 Z y 6=i2 = ;1. . A B| -
N=
C = f x + iy : x y 2 R g, N Z Q R C
,
,
R
,
C
. .
+ Z , Q+, R+
x 2 B g. . A, B, A\B | f x : x 2 A x 2 B g. A B. A ; B = f x : x 2 A x 2 B g. = ( ) | A B = f (x y) : x 2 A y 2 B g | . (1596{1650). (x y). , A, }(A) = f B : B A g. , A }(A). I F = f Ai g, i2 I Ai F. I| n , n F = f A1 : : : An g. , , i2I Ai = f x : ,
fx : x 2 A
A B
37
x 2 Ai
Y= An ,
n Ai = A1
i g \i2I Ai = f x : x 2 Ai An = f (x1 : : : xn) : xi 2 Ai g: n.
i g.
1 i n
n-
. b. c.
$
, A 2 $, A 6= . A 2 $ B 2 $, $. , , , .
2.1.1.
,
: A = B, S S|
S A\B = .
A = A1 = An. $, R .
S
f1 2 3 4 5 6g, .
. , f f1 2 3g f4 5 6g g | . R , A, B, C $.
S
A \ (B C) = (A \ B) (A \ C).
2.1.2.
. , A \ (B C) (A \ B) (A \ C). x 2 A \ (B C). x 2 A (x 2 B x 2 C). x 2 B, x 2 A \ B. x 2 C, x 2 A \ C. x (A \ B) (A \ C). .
2.1.2., A A, . . R , , x < y, LT = f (x y) : y ; x R A A A.Z
-
Z
Z
g.
38
A
n -
2.1.3.:
E A A, ).
a. (x x) 2 E x2A( b. (x y) 2 E (y x) 2 E ( c. (x y) 2 E (y z) 2 E). E x Ey( (x y) 2 E x2A x = f y 2 A : y x g, x. ), x y,
(x z) 2 E ( : x
y.-
y2x y 2 x, y = x,
. , ,
A= = f x : x 2 A g. A (x0 y0),
( A.
)
., ,
xy0 = x0 y.Q
Z Z
x=y = x0=y0 , A , A,
(x y) , (;3 ;9) (2 6) (;1 2) (3 ;6). . Z Z= Q. , x=y, x 2 Z y 2 Z , , xy0 = x0y. , Q | . . E| A. , A=E = f a : a 2 A g $| A,
2.1.4.
39
E , E ,.
A. A. a2A
A=E a 2 a. , .
,
x2A a b (x a) 2 E (x b) 2 E (a x) 2 E (x b) 2 E, (a b) 2 E a = b. b E$ A, , $ (a b) 2 E$ .) $, A=E$ . x2a x 2 $a ,
A. , x2a x2b
a
a = b.
, a $a , -
$a = a 2 A=E$ . (x a) 2 E$ , x 2 $a x2a , .
$. ( A=E$ = $ a 2 A. $a E$ , E$ , $
, ,
a
a $a .
(x a) 2 E$ a, $a a,
$a
a.
A=E$ . A=E$ $ , -
A. . m > 1. Z b m a, : b ; a = mq q2Z , b m a, m. : ;4 5 16, 91 7 0 1087 2 1. a| a = f a + mq : q 2 Zg, a = a + mZ. m = 5, 0 = f : : : ;10 ;5 0 5 10 : : : g, 1 = f : : : ;9 ;4 1 6 11 : :: g
,
40
0 1 2 3 4,, m .
, , .
Z
Z 5= =
f 0 1 2 3 4 g.::: m; 1g
,
Z m = f0 =
2.1.3.. B A
2.1.5.
: x2A y 2 B, , (x y) 2 f. x 2 A f(x), x f x| y 2 B, x. f A B f f: A ! B A ! B. f(A) = f f(x) : x 2 A g ( B) A f. y2B f ;1 (y) = f x 2 A : f(x) = y g ( A) y. y| A, y. ( . , f ;1 A B, .) ( , , | ), , ( ), (homo = , morphism = ). , f: A ! B, A B| , , i. f(a + b) = f(a) + f(b), ii. f(a b) = f(a) f(b), iii. f(1A ) = 1B . 1A 1B | ( . 2.1.13) A B. ( , iii , .) f
x2A f A B,
A B| B ,
. y 2 B.
A A B, . .
A B
41
( B A
),
) f: A ! B f(A) = B ( , , A, B
f(x) 6= f(x0 ) ( B). A = B.
, A). , . A B|
( f x 6= x0
-
-
.
. x 2 A. id: A ! A , p1 : A . , s: A ! A= , A, , s;1 (4) = 4, 5 , -
1. 2. 3.s(x) = x, , .Z .
id B!A
. p;1(x) 1 A (
A id(x) = x x 2 A? B. p1 (x y)] = x. ) x A= .
x, A=
, s5 : Z! Z 5 = , 4 = f : : : ;6 ;1 4 9 :: : g, ,|
4. 5.
n + 1. +
succ: N ! N succ(n) = .( ?) +: Z Z! Z +(x y) = x + y | , . +;1 (0)? ? , | . S . x, y S x y = y x, .
42
x (y z) = (x y) z, Z + S . , 2.1.7) , , S n2N.
xyz S . , , succ: N ! N,N
, .
.
;
,
S S( . succ(n) = n + 1, .
f: S ! S , , -
.
2.1.6.S.
f1 2 ::: ng. f: S ! S |
2.1.7.s 2 S.
.
S| f
f
.
, , f(t) = s g: S ! S g(s) = t. g(s1 ) = g(s2 ), s1 = f g(s1 )] = f g(s2 )] = s2 , g , S ,g . f , s1 6= s2 S, , f(s1 ) = f(s2 ). , g , , g(t1 ) = s1 g(t2 ) = s2 . t1 t2 S, t1 = f g(t1 )] = f(s1 ) = f(s2 ) = f g(t2 )] = t2 , g(t1 ) 6= g(t2 ). f g: A ! B , f = g, f(x) = g(x) x 2 A. f: A ! B g: B ! C | , f(x) 2 B x2A , , g f(x)] . f g g f: A ! C, g f(x) = g f(x)] x 2 A. g f : f, g. = . 5| +5 : Z Z! Z 5, +5 (x y) = s5 (x + y). t 2 S,
43
+ .
: +5 = s5 +. , ).
s5
2.1.8 ( f: A ! B, g: B ! C h: C ! D h (g f) = (h g) f. . x2A h (g hfg f(x)]g = h g f(x)] = (h g) f(x). f: A ! B | , f x g, f ;1 : B ! A y y 2 B. f ;1 f: A ! A f f ;1 : B ! B f ;1 f(x) = x f f ;1 (y) = y. f f ;1 = idB , idA idB ..
f)(x) = h g f(x)] = y2B f ;1 (y) , f(x) = y. , x, , f(x) = y, f f ;1 , , , f ;1 f = idA ,
a. b. c.
g ;1 .
fg fg fg
2.1.9.
f: A ! B g: B ! C g f . , g f . , g f (g f);1 = f ;1 , . a,
g B C, g(b) = c. , f , a 2 A, , f(a) = b. g f(a) = g f(a)] = g(b) = c, g f . f: A ! B. A : x x0 2 A , x x0 , f(x) = f(x0 ). , . i: A= ! B i(x) = f(x) x. x c2C b 2 B, ,.
44
i i f(x) = f(x0 ), .
x x0 2 x, .
i
s: A ! A= | . f A!B s& %i A=
i
x = x0 , i(x0 ) = f(x0 ).
,
.
x0
x
. , -
f: A ! B |
2.1.10 (
.
).
f: Z ! N, f(x) = x2, x 2 Z . f = i s, | s: Z! Z ()2 = f0g f x = fx ;xg : x 2 Z g, = | i: Z ()2! N. = 2.1.10 , f A= . ( , ), , , . A, . , A|id
f = i s.
f
s
i,
-
.
2.1.11.
B,
i: A= ! f(A), , . ,
f(A) | f(A).
-
A A 2.1.8
f Bij(A) = f A ! A, f |
g.Bij (A).
,
f ;1
Bij(A),
f
. .
,
2.1.9
Bij(A),
45
2.1.12 (f
f A ! A, f | g. : a. f, g 2 Bij (A) f g 2 Bij (A). b. h (f g) = (h f) g f, g, h 2 Bij (A). c. id f = f id = f f 2 Bij (A). ;1 2 Bij (A) f ;1 f = f f ;1 = id. d. f 2 Bij (A), f a 2.1.12 , Bij (A) , b| , c| Bij (A) d| Bij (A)Bij (A).
.
).
A|
A, Bij (A) =
-
| n ( n. n
, (
,
,
.
. ), n = f1 ::: ng | ~ . x . ,
, Bij (A) 1
). Sn = Bij (~ ) n . Sn n! 1=x .
|
a. R(
,
2.1.13.,
(R + ) x+y = y+x , , 0R ,
: + x y 2 R). 0. ) 1R
b. R, 1.
(
46
c.
xy + xz (y + z)x = yx + zx Z Q, R, C , .R+ Z 5g ;f
x y z 2 R. , + ,Z;f5g
: x(y + z) = : Z;f5g a, b 2 Z;f5g , 2+3 = 5. , 0, 1 .-
N,
,
5. , , ,Z; f5g Z; f5g
a+b , .Z ,
|
. , ab = ba. R| . . ,
,
n > 0. 0 n > 0, .
0 6= 1 + 1 + , Z Q, R C . , 0 = 1 + 1 + + 1 (n
+ 1 (n R ),
)
, a 6= 0 ab = 0 u 6= 0 R , . . uv = 1 . R b 6= 0 (b , , -
2.1.14.). v 2 R (v = u;1). ,
.
(Z m + ) = m . . . ,
, m = 8 2, 4 2 Z 8, 2 4 = 0 = . , m=5 , 2, 3 4 |
Z m =
,2 4
3, 2 4
Z 5 =
47
, , . ( , .Z
,
-
2.1.15.1 ;1. )| ,
2.1.16.(F + ) | : . F . .C
,
, +.
,
, -
a. F | b. c., ,
R Q
.
. (GF ).
, 2|
,
f0 1g
GF(2) q +1 (k
GF (q) ) ,
.
-
X1 i k
1 = 1+1+
k = 1 2 3 : : :. . k0 k00 , k0 < k00, , , 1=0
, -
X
X1 i k00
1 i k
1:
48
,
P
1 i k ;k00
0
1 = 0,
, ,
,
GF (2)
1 i 1 = 0. GF(q). , 1 + 1 = 0.
P
-
2.2| . . , k0 | ..
, -
2.2.1.(Childs, 1979). . k0 , k0, S| p(k) | p(j) . , , p(k) , S k0 ,
., j , k
, . , k. S S S,
, S
S ,
, p(k0 ) k0. ,
k0, p(n)
k0
n k0 p(n) ,S S .
,
, p(n + 1) , n+1 , n, , n+1 . , p(n + 1) S . ( . .3 . 2.2.1) . S.
p(n+1)
49
, . . , a b a b( a j b), c, , b = a c. , 7 j 28, 28 = 7 4 28 = (;7)(;4). a a j 0, 1 j a a j a. . | , . 2.2.1 ( ). a b ( ) q r, , a = b q + r, 0 r < jbj. . a ; kb, k , .. : : : a ; 3b a ; 2b a ; b a a + b a + 2b a + 3b : : : : r, q k. ( r , f a ; kb g , , .) r = a ; qb 0: , a = b q1 + r1 0 r1 < jbj r1 6= r. r1 < r, 0 < r ; r1 < jbj k0 , S bj(r ; r1), r ; r1 = (q1 ; q)b d>0 0 < r ; r1 < jbj.
:
S|
2.2.2.c j d.
a
b
. a b,
a. d j a d j b. b. c j a c j b,
50
gcd(a b) ,
(a b). ,
a :
b . d0 | b d j d0 d0 j d, . , (12 30) =
b
-
, d0 = d, (12 ;30) = (;12 30) = (;12 ;30) = 6. ,
2.2.3 (ax + by. , ax + by, d>0 d.
gcd). d| , d = ax0 + by0 , .( , d
, . a b x y, x0 , y0
,
(a b) = -
d
b
a , d b. b = d q + r, 0 < r < d, , r = b ; dq = b ; (ax0 + by0 )q = a(;qx0 ) + b(1 ; qy0 ), d. , x y , , . , 6 = (12 ;30) = 12(3) + (;30)(1) = , 12(;2) + (;30)(;1). ,2.2.3. ( ). , ax + by = c, d = (a b). x y , n.
2.2.1, .) 2.2.2. a.
, , , ,
a b , x0 ;
2.2.4.d j c. b. x0, y0 | n(b=d), y0 + n(a=d)
a.
,
51
, b , x y| , , ax + by = c , d j a d j b, d j c. , d j c, . . c = dk k. 2.2.3 s, t, , d = as + bt. k, c = dk = a(sk) + b(tk), , x = sk y = tk ax+ by = c. : 12x;30y = 84 , (12 ;30) = 6j84. x = 2, y = ;2, x = 2 + 5n, y = ;2 + 2n. a b , (a b) = 1. 2.2.3 s, t, , as + bt = 1. . 2.2.5. a b , d = (a b). a=d b=d . . 2.2.3 s, t, , d = as + bt. d, 1 = (a=d)s + (b=d)t, (a=d b=d) = 1. 2.2.6. a, b c | d = (a b). a bc, a=d c. . a j bc, (a=d) j (b=d)c. 2.2.5 (a=d b=d) = 1, 2.2.3 s, t, , (a=d)s + (b=d)t = 1 c, c(a=d)s+c(b=d)t = c. a=d c(a=d)s c(b=d)t, c. 2.2.7. a b| . m>0 a b,.
.
. a j m b j m. b. a j c b j c,a b].
m j c. a b m. lcm). a b| , lcm(a b)
2.2.8 ( a b] = jabj=(a b).
b
52
(a=d)b u(a=d)
d = (a b) , s, t. , m j n, 2.2.7
.
a b b=d a=d a b a j bt,
, a b. n = as = bt u.
.
,
(ab)=d = a(b=d) = , m = jabj=d | n| 2.2.6 (a=d) j t t = n = bt = u(ab)=d .
b
a b.
m| , , m
2.2.2.b a = bq +r 0 r < jbj. , q a . r, a , . 2.2.1 b
.1
QUO(a b) MOD(a b),
. : a = bq + r d a b, d j r = a ; bq ( . 3 . 2.2.1). , d= gcd( b) , gcd(a b) = gcd b MOD(a b)]. , (a 0) = jaj a gcd(0 0) . , a b, : a0 = a a1 = b a0 = a1q1 + a2 0 < a2 < ja1j a1 = a2q2 + a3 0 < a3 < a2 ::: ak;2 = ak;1qk;1 + ak 0 < ak < ak;1 ak;1 = ak qk + 0: a2 > a3 > . >0 , , ak
ja1j >
53
,
, ,
(a0 a1) = (a1 a2) = x := x1 , y := y1 .) (Euclidean Algorithm)
x := y (x y) := (x1 y1)
= (ak 0) = ak a b . ( x y
: a b 6= 0. : d = gcd(a b). 1. ] (a0 a1) := (a b). 2. ] a1 6= 0 (a0 a1) := a1 MOD(a0 a1)]. 3. ] d := a0. a = 342 b = 612. (342 612) = (612 342) = (342 270) = (270 72) = (72 54) = (54 18) = (18 0) , d = 18. EA. , a b. 1 3 1, , 2. 2 n , n , , , a b, . 1.2 , tIDIV (a b) = OfL(b) L(a) ; L(b) + 1]g. , tIDIV(a b) . , tIDIV (a b) = OfL(b) L(a) ; L(b) + 1]g: ( , a b), , 5( n, n , -
EA.
) n 5 L(b).
54
, tEA (a b) = ( )( 5 L(b)] fL(b) L(a) ; L(b)+1]g, tEA (a b) = OfL(b)2 L(a) ; L(b) + 1]g: )
(Lame, 1844) (). ..
,
, , ,
,
-
,
. -
, ( . , , -
2) 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 : :: . ( 1| , f1 1 2 3 5 8g f1 2 3 5 8g.) , , , . , t1 k+1 , t1 t2 = t 1 + t 0 , 10k < t1 < 2 10k k (1=2)10k < t0 < 10k (3=2)10k < t2 < 3 10k : : (5=2)10k fk;1 , ( fk = fk;1 + fk;2 fk;1 ) rh = rh;1 + rh;2 fk;2 > rh;2. . , (fk;1 fk;2) fk = rh > rh;1 > fk;1 fk > rh > rh;1 = fk;1 . , 1, ( ), , q > 1, . . ri = 2 ri;1 +ri;2 ( ri 2fj ; fj +1 > 0 ,
,
,
, rn rn ;1 : : : r00 0
.
fj +1 fj |
q > 1).
ri ; 2ri;1 > 0
,
2(fj ; ri;1) ; (fj +1 ; ri ) > 0 , fj > ri;1. ri;1 , fj ;1 , (fj fj ;1) . , ri;1 > fj ;1 , fj +1 = 2fj ;1 + fj ;2 , ri = 2ri;1 + ri;2 ri < fj +1 , fj ;2 > ri;2, . . (fj ;1 fj ;2) . ,
56
,
, , , , r1 .
, : : : r1 r0 : : : f1 f0, , r0. , f0 = 1, f1 = 2, r1 r0 = 1 .
.
,0 0
-
rn , rn ;1, fn fn;1, -
b
fn+1 , , -
( .
, 1979 Wilf, 1986). a b 1,
2.2.9..
, MOD(a b) b ; 1. (a + 1)=2. b;1 (a + 1)=2 ; 1 = (a ; 1)=2. ii. b > (a + 1)=2. a;b a ; (a + 1)=2 = (a ; 1)=2. ). = max(a b).
MOD(a b) (a ; 1)=2. MOD(a b) = a ;ba=bc b a ; b , MOD(a b) min(a ; b b ; 1),a;b b;1
i. b
MOD(a b) MOD(a b)
: b;1
a;b < -
2.2.10 (
a
b |
,
b2 log2 M c + 1..
a
b, , a b.
a0, a1 , : : :, ak , a0 = a, a1 = b ai = MOD(ai;2 ai;1), i 2. 2.2.9 ai (ai;2 ; 1)=2 ai;2=2. i a2i a0=2i a2i+1 a1 =2i, i 0, ;bk=2cM. ak 2 , ak < 1 , 2;bk=2cM < 1, . . k > 2 log2 M.
57
89) (21 13). , 5 2 = 10 5. 1849), (144 89) ,
.
, 15, 2.2.10, 9 b| ,
, | 9 , (21 13) |
(144, -
, 6=$2 0:60793. ( ( , , 1988 Bradley, 1970 Collins, 1974 Knuth, 1969 Lipson, 1981 Motzkin, 1949 Schroeder, 1986).)
a
(Dirichlet, gcd(a b) = 1,
2.2.3.( 2.2.3). , a b gcd(a b) = ax + by . 270 = 612 ; 342 72 = 342 ; 270 54 = 270 ; 72 3 18 = 72 ; 54 , 612 ,
a = 612 b = 342 612 = 342 1 + 270 342 = 270 1 + 72 270 = 72 3 + 54 72 = 54 1 + 18 54 = 18 3 + 0 342. , 18 |
18 = 72 ; 54 = (342 ; 270) ; (270 ; 3 72) = 342 ; (612 ; 342)] ; 8612 ; 342) ; 3 (342 ; 270)] = 342 ; (612 ; 342)] ; f(612 ; 342) ; 3 342 ; (612 ; 342)]g = 9 342 + (;5) 612 . . 18 = 9 342 + (;5) 612, , . , x y , , . ,-
58
,
ai ( . 2.2.2)
axi + byi .
-
axi + byi xi yi . , x0 = 1, y0 = 0 x1 = 0, y1 = 1. i, ai = axi + byi = ai;2 ; ai;1qi;1 = (axi;2 + byi;2) ; (axi;1 + byi;1)qi;1 = a(xi;2 ; xi;1qi;1) + b(yi;2 ; yi;1 qi;1), xi yi : qi;1 = QUO(ai;2 ai;1) ai = ai;2 ; ai;1qi;1 xi = xi;2 ; xi;1qi;1 yi = yi;2 ; yi;1 qi;1: , ai MOD(ai;2 ai;1), , ai , xi yi . . XEA. (Extended Euclidean Algorithm) : a b 6= 0. : d, x, y, , d = gcd(a b) = ax + by. 1. ] (a0 a1) := (a b) (x0 x1) := (1 0) (y0 y1 ) := (0 1). 2. ] a1 6= 0, f q := QUO(a0 a1) (a0 a1) := (a1 a0 ; a1 q) (x0 x1) := (x1 x0 ; x1 q) (y0 y1) := (y1 y0 ; y1 q) g. 3. ] (d x y) := (a0 x0 y0). .
a0 = a, a1 = b, a2 = a0 ; a1q1 , a3 = a1 ; a2q2 , ::: ai = ai;2 ; ai;1qi;1, ::: ak = ak;2 ; ak;1qk;1, 0 = ak;1 ; ak qk , |
a0 = ax0 + by0, a1 = ax1 + by1, a2 = ax2 + by2, a3 = ax3 + by3, ::: ai = axi + byi , ::: ak = axk + byk , 0 = axk+1 + byk+1. ,
59
. a = 342, b = 612, : q a0 a1 x0 x1 y0 y1 1 0 0 1 0 612 342 0 1 1 0 1 342 270 1 ;1 0 1 1 270 72 ;1 2 1 ;1 3 72 54 2 ;7 ;1 4 1 54 18 ;7 9 4 ;5 3 18 0 9 ;34 ;5 19 , a0 = ax0 + by0 d = 18, x = 9, y = ;5 342 (9) + 612 (;5). 0 1 2 3 4 5 6
EXA
EA,
; 342 612
. : 18 =
2.2.4.. .7 , , . -
(Olds, 1963 Richards, 1981). a1
a0=a1 , , . . (a0 a1) = 1 a1 > 0. a0, , a 0 = a 1 c 0 + a2 0 < a2 < a1, a 1 = a 2 c 1 + a3 0 < a3 < a2, a2 = a3c2 + a4 0 < a4 < a3, ::: ak;2 = ak;1ck;2 + ak 0 < ak < ak;1, ak;1 = ak ck;1. , . 2.2.2 q1 : : : qk c0 : : : ck;1. ai =ai+1 i 0 i k ; 1, ii+1
i = ci +
1
0 i k;2
k;1 = ck;1:
600 = c0 + 1= c0 + 1=(c1 + 1= 1 ). 1 0 = c0 + 1
c1 + 1= 2, , 1 1 ck;2 + c 1 . , c1 , 8=5. , a1 > 0), c0 . , 1 :
0
=
a0 = a1
c1 + .
..
+
k;1
| , (c0 c1 : : : ck;1) 8=5 = (1 1 1 2). ,
a0=a1 (
.
a0
ci -
0 < a2 < ja1j, c2 : : : ck;1. , .
.
.
-
,
, 8=5 = (1 1 1 1 1). -
a0 = (c c : : : c c ) = (c c : : : c c ; 1 1): 0 1 k;2 k;1 0 1 k;2 k;1 a1 . , c1 c2 : : : ck;1 , . , ;8=5 (;2 2 2) (;1 ;1 ;1 ;2). , , 8=5, . .
2.2.11.c] =
. c] c
bcc dce
c 0 c < 0: ). (c0 c1 : : : cm ) = (d0 d1, m = n ci = di i = 0 1 : : : n.
: : : dn)
2.2.12 (
cm > 1 dn > 1,
61
. . si = (ci ci+1 : : : cm ) ti = (di di+1 : : : dn). , si = (ci ci+1 , : : :, cm ) = ci + 1=si+1 ti = (di di+1, : : :, dn ) = di + 1=ti+1. , si > ci, si > 1 i = 1 2 : : : m ; 1, sm = cm > 1 ti > di , ti > 1 i = 1, 2 : : :, n ; 1, tn = dn > 1 , ci = si] di = ti] i . s0 = t0 , , , c0 = s0 ] = t0] = d0. s1 , t1 1=s1 = s0 ; c0 = t0 ; d0 = 1=t1, , s1 = t1 c1 = s1 ] = t1] = d1. : , si = ti ci = di, , si+1 = ti+1 ci+1 = di+1. ,m n. , , m < n. , sm = tm , sm = cm tm > dm . cm = dm . 2.2.13. , , , , . .
c0 +1=(c1 c2, : : :, cm ). 2.2.12. ? 2.2.14, : : :), , ,
(c0 c1 , : : :, cm ) = . (c0 c1, : : :, cn , : ci = b i c . ci ,
2.2.14.ci0
i+1 = 1=( i ; ci ),
=
i 0.
ci > 0 ,
i,
,
p;2 = 0 p;1 = 1 pi = cipi;1 + pi;2 i 0 q;2 = 0 q;1 = 1 qi = ci qi;1 + qi;2 i 0
62
(c0 c1 : : : cn) rn = pn =qn, (pn qn) = 1, .
qn
n-
-
a.
n > 0, = (c0 c1, : : :, cn;1, n ),
: n = (cn cn+1, : : :), n 0,
= pn;1 n + pn;2 qn;1 n + qn;2
b.
= c0 +
1
c1 + .
..
1 cn;1 + 11
, 1
n 1
+
c.
pn = c0 + qn
n
c1 + .
..
1 cn;1 + c1 1
,
n 0.
+ ,
, d| , 2.2.14 . ,
n
,
. (p (p d)=q, q2 x2 ; 2pqx+(p2 ; d), .) , x(
-
,
1, . 1, .
x1 ,
). c0 , , x, x x = c0 + 1=x1, 0 < 1=x1 < , x x1 = 1=(x ; c0) > 1 c1 , , x1 x1 = c1 + 1=x2, 0 < 1=x2 < 1, c1 x2 = 1=(x1 ; c1 ) > 1 , x
63
x = (c0 c1 , : : :),
, ,
x . n 1. n 1. n 2.
.
. = (c0 c1,
: : :),
a. b. c. d.
rn = pn=qn | n: pnqn;1 ; pn;1qn = (;1)n;1, rn ; rn;1 = (;1)n;1=qnqn;1, rn ; rn;2 = (;1)n;2cn =qnqn;2, n , n , , r2n;1,.
2.2.15.
n, rn , n r2n 2,
.
. n = 1, 2.2.14, p1 q0 ; p0q1 = (c1 p0 + p;1) 1 ; c0c1 = (c1c0 +1);c1 c0 = 1 , p0 = c0 , q0 = 1 q1 = c1q0 q0. , n = k, . . pk qk;1 ; pk;1qk = (;1)k;1. , k + 1. 2.2.14, pk+1 qk ; pk qk+1 = (ck+1 pk + pk;1)qk ; pk (ck+1qk ; qk;1) = ;(pk qk;1 ; pk;1qk ) = ;(;1)k;1 = (;1)k . b. pn qn;1 ; pn;1qn = (;1)n;1 qn qn;1, pn=qn ; pn;1=qn;1 = (;1)n;1=qnqn;1. , rn = pn=qn. c. rn ; rn;2 = pn=qn ; pn;2=qn;2 = (pn qn;2 ; pn;2qn )=qnqn;2. pn qn ( 2.2.14) cnpn;1 + pn;2 cnqn;1 + qn;2 , pnqn;2 ; pn;2qn = (cnpn;1 + pn;2)qn;2 ; pn;2(cnqn;1 + qn;2) = cn (pn;1qn;2 ; pn;2qn;1) = (;1)n;2cn = (;1)n cn. d. . .b c , r2n < r2n+2, r2n+1 < r2n;1 r2n < r2n;1, qn n 1. , r0 < r2 < ::: r1 > r3 > r5 > : : : .
a.
64
, n, , , p;1 q;1, , n. 2.2.14 p;2 q;2 .
r1 r0 . rn ; rn;1 , 2.2.15 pn
qn qn,
p;1 = 0 p0 = 1 pi = cipi;1 + pi;2 i 1 q;1 = 0 q0 = 1 qi = ci qi;1 + qi;2 i 1 c0 , .b c .7 . c1 . , .a . pn qn;1 ; pn;1qn = (;1)n n 0 n 2 n 3
144=89 , 2.2.14 , . 0 = 144=89, c0 = b144=89c = 1, p;2 = 0, p;1 = 1, q;2 = 1 q;1 = 0 , pi = cipi;1 + pi;2 qi = ci qi;1 + qi;2, p0 = c0 q0 = 1 p0 =q0 = 1 | 144=89, . = 1=( 0 ; c0) = 89=55, c1 = b89=55c = 1 , 1 , p1 = 2 q1 = 1 p1 =q1 = 2 | 144=89, . = 1=( 1 ; c1) = 55=34, c2 = 1, 2 p2 = 3, q2 = 2, p2 =q2 = 3=2 | 144=89, , p0 =q0 < p2 =q2 p0=q0 < p2 =q2 < p1=q1. , = 1=( 2 ; c2 ) = 34=21, c3 = 1, p3 = 5, q3 = 3 3 p3 =q3 = 5=3 | 144=89, p3=q3 < p1 =q1 p0 =q0 < p2 =q2 < p3 =q3 < p1 =q1. . ,
.
65
c1 , : : :, cn;1 , n ) | : : :), n 0.
2.2.16..
|
,
, :
n = (cn cn+1 ,
= (c0
a.
,
p=q, -
2 b. 1=(2qn+1qn) < j ; pn =qnj < 1=(qn+1qn) < 1=qn, n 0. c. (p q) = 1, , j ; p=qj < 1=q2 .
a.
2.2.14 = (pn;1 n + pn;2)=(qn;1 n + qn;2 ), (qn;1 n +qn;2) = (pn;1 n +pn;2), , , n ( qn;1 ; pn;1) = ;( qn;2 ; pn;2) = ;qn;2( pn;2=qn;2). nqn;1 , j ; pn;1=qn;1j = jqn;2= nqn;1j j pn;2=qn;2j. n 1 qn n>1
.
;-
;
, qn;1 > qn;2 0 < jqn;2= n qn;1j < 1, , j ; pn;1=qn;1j < j ; pn;2=qn;2j, j ; rn;1j < j ; rn;2j n 2. b. .b 2.2.15 jrn+1 ; rnj = 1=qn+1qn, n 1 , , rn+1 , rn, , , 1=2qn+1qn < j ; pn=qnj < 2 2 1=qn+1qn < 1=qn, n 0, 1=qn+1qn < 1=qn, qn+1 > qn. ( . . 2.2.1.);;;;;;;;;;;;;;; 1=q +1q ;;;;;;;;;;;;;;;! ;;;; 1=2q +1q ;;;;!n n n n
rn =pn =qn
. 2.2.1., . b.
m
rn+1 =pn+1 =qn+1
b
2.2.16. pn=qn, , -
c., .
2 jx ; pn=qnj < 1=qn
66
.
2, 1, 3, 1, 14, 2 : : :). x , , , x. $ = 0 = 3:14159. 3:14159 = 3 + 0:14159 1 0:14159 = 7 + 0:06264 1 0:06264 = 15 + 0:96424 1 0:96424 = 1 + 0:03708 ::: , 3:14159. r2 = 333=106 r3 = 355=113. j ; p=qj < 1=q2. r2 $ (2 1 2 1 1 4 1 1 6 1 1 8 1 1 :::) . , (Lang, Trotter, 1972) , ,
.
,
$ $ = (3 7, 15, 1, 292, 1, 1, 1, , , ,1 = 0:14159 2 = 0:06264 3 = 0:96424 4 = 0:03708
x,
-
,
2.2.14,;1
c0 = 3 c1 = 7 c2 = 15 c3 = 1
;1 ;1 ;1
$ = (3 7 15 1 : ::). $
,
-
, $: r0 = 3=1, r1 = 22=7, j3:14159 ; 333=106j < 1=1062, 0:00007 < 0:000089. : e : e= $ e . , j, 1 n-e
cn
2 log2 (j + 1) : j(j + 2)
67
, , ,
0:41. , , , ( . = (2 3 2 3 2 3 :: :). 2.2.14 , . , ,
,
cn = 1, 2.2.14)
? = (2 3 2) = (2 3 ). 3 2 ; 6 ; 2 = 0. ,
= 2 = 4 : : :, = 2 + 1=(3 + 1= ), , p = (3 + 15)=3 | .
.p p
= 7
p
2.2.14, c0 =2 c1 =1 c2 =1 c3 =1 c4 =4 p 7 = (2
7=2+( 7;2) p p p 1=( 7;2)=( 7+2)=3=1+( 7;1)=3 p p p 3=( 7;1)=( 7+1)=2=1+( 7;1)=2 p p p 2=( 7;1)=( 7+1)=3=1+( 7;2)=3 p p p 3=( 7;2)= 7+2=4+( 7;2) , 5= 1
1 1 1 4 1 1 1 4 : : :).
p ;1 7;2) p ;1 2 = ( 7;1)=3] p ;1 3 = ( 7;1)=2] p ;1 4 = ( 7;2)=3] p ;1 5 =( 7;2) :1 =(
2.3, (
, (Dickson, 1952). n > 0, a b, , : , ,| ,
, , , ab = n. ,
, :
). ,
-
, , ,|
a b, .
, .
-
68
2.3.1., . a. , . , a 6= 1 ui > 1 i = 1 ::: r , , 276 = 12 23 | . . 1 a. a 6= 1 a = bc, , b a 6= 1 , .. , . c . , ). . , 13 ;7 , .. 1 .-
2.3.1 (. a = u1 ur ,.
,
,
, a > 0. a , a = bc, b c 1 1 < b c < a. b c , , b c . , . -
, pja
, 1008 = 2 2 2 2 3 3 7. , .. 1008 ? p j b. , a, b
. , .
p j ab, ,
p>1 , .
,
2.3.2.
p>1 .
69
b. ( . .2 p = ab = apc, , p j ab. p
,
a|
.
x y, , abx + pby = b, , p j ab
p>1 ajp . p = ab p ,pja p j b. p j a, a = p . 2.2.1). p j b, b = pc c 1 = ac a = 1. , ,p p , , , p>1 a, (a p) = 1, p , 1 p. 2.2.3 ax + py = 1. b, , p , p j b. , p a, p j b, ,p ..
,
2.3.3 (..
a 6= 1 , ur = v1 ,
). , vs , -
u1 j2 ,
, , u1 j vj1
a = u1
a ui vj . u1 j v1 vs , j1. vj1 , u1 = vj1 . , , u2 vj , , u2 = vj2 . , r=s ui = vi i. 2.3.3 , . , , , -
2.3.3
E , 8=2 4|
,
,
. E, 2, 4, 6, 8 : : : . . , E, 14 | , , ,Z
, 840 840 = 2 14 30 = 6 10 14, . 2.3.3
70
.
, a 6= 1 ,
2.3.1 2.3.3 ). a :
, ( -
a = pe1 pe2 1 2 pi | . d, , a b.
pek k ei | , . a b, ,
ei m n, ,
a = pe1 pe2 pek 1 2 k 0 fi 0. ,
b = pf1 pf2 pfk 1 2 k m j pn , p a bmin( pk ek fk )
d = (a b) = pmin(e1 f1 ) pmin(e2 f2 ) 1 2 . , min(i j) , a b] = pmax(e1 f1 ) pmax(e2 f2 ) 1 2 max(i j) . ,
, a b pmax(ek fk ) k ,e pkk +fk
-
-
jabj = pe1 +f1 pe2 +f2 1 2
min(a b) + max(a b) = a + b, 2.2.8: a b] = jabj=(a b). 2.3.4. ,
2.3.4.a b, ,
p.p
71
2| , 2 , .
. ,
, 2 = a=b, (a b) = 1. , ).
.
2b2 = a2 | 2.3.5 (.
,
p. , n , 2 + 1 = 3, 2 3 + 1 = 7, 2 3 5 + 1 = 31, 2 3 5 7 + 1 = 211, 2 3 5 7 11+1 = 2311. 2 3 5 7 11 13+1 = 30031 = 59 509.) p pi , p p1 p2 pm , p n = p1 p2 pm + 1, , .. , . p . ( n
n = p1 p2
pm + 1,
. p1 p2 : : : pm . ,
,
2.3.6.n
. . (n + 1)! + 2, (n + 1)! + 3, : : :, (n + 1)! + (n + 1), k! = 1 2 k. , , (n + 1)!+ i i, 2 i n + 1. , m n , (m n) = 1 , m n , , , 9 14. n>0 (n) m, , m n (m n) = 1 . , p (p) = p ; 1. , (5) = 4, 1, 2, 3 4 5. , (n) n n = pe1 pe2 pek . 1 2 k m, m n, pi,
.
(.. ,
)
72
n. , , .
, n n=pj , j = 1 2 : : : k, pj . , m. , ,
m n ; n=p1 ; n=p2 ; ; n=pk , p1 p2 , . , n=(p1p2 )+n=(p1p3 )+ +n=(pk;1pk ). , , , p1 p2p3 , , , n n n (n) = n ; p ; p ; ; p 1 2 k + p np + p n + + p n p 1 2 1 p3 k;1 k n : k + + (;1) p p p 1 2 k ,1)
,
m
,
,
(n) = n 1 ; p1 1
1 ; p1
2
1 ; p1 :k
.,
(60), : 60 = 22 3 5.
,
60
(60) = 60(1 ; 1=2)(1 ; 1=3)(1 ; 1=5) = 16: . .1)
.
, , :
. (pk ; 1)pek ;1: k ,
(n) = (p1 ; 1)pe1;1 (p2 ; 1)pe2 ;1 1 2 e .|. .
73
.
2,
2 p1 : : : pi. , i+1,
n. ,
n, 2 , 3, x > pi , .( , , ?) n. . , i-
, i -
pi+1, .
, 3 5 23 25 n 43 45 n = 32, n, 7, | 11, , n, 2 21 = 41 3,
, 7 1947).)
60, 2: 7 = 11 13 15 17 19 9 = 27 29 31 33 35 37 39 = = n = 47 49 51 53 55 57 59 | = n = =, 5, 52, | 2, | 7 |. 112 > 60, , , 60. ( . (Dudley, 1983) (Mills, , , , , n , pi , , , . , .) p2 n i -
pn., (, i. . , , n , . ,
pn, ,133951979 .,
, 106684 M, M|
244497 ; 1,
74
, | 16
, , , 215 ; 1 = 32767 231 ; 1 = 2147483647
. 32
,
, . L, L + 1, : : : M, , , , n , -
p
M, . . M (L M), ,
. ,
(L M)
n. ((Erdoes, 1949 Goldstein, 1973 Levinson, 1969), . 3.) $(x) x , , , $(60) = 17. , limx!1 $(x)=(x= lnx) = 1. x x= ln x $(x) . , , ln M . , , n lnM, n p . , p M= ln M. n = 10, 1610 ln 225 104, , , 200. p 15 p 15 2 = ln 2 35 . p 15 3210 ln 231 214, $( 2 ) = 42. 500. p 31 p 31 p 31 | 2 = ln 2 4314. $( 2 ) = 4691. .
n
GENPR.: : A
(Generate Primes) k m k m| p1 < p2 < : : : < pr m m + 2k ; 2]. 3. ,
75
j, , dj (m + 2j ; 2) m + 2j ; 2 3] r := MOD(m d) j := 1 r>0 r , j := j + d ; r=2 r>0 r , m d, j := j + d. j := j + (d ; r)=2 4. ] i := j, j +d, j +2d : : : j>k A(j) := 0. 5. d] MOD(d 6) = 1, d := d+4, d := d+2 2. 6. ] i := k, k ; 1, : : : 1 f A(i) = 1, m + 2i ; 2 g . GENPR. , , O(1). 1 6 , k O(k). 2 5 , , pn, n = m + 2k ; 2, 2, . , 4 k , 2, 3 5 | p . O(k n). p p , O(k +k n) = O(k n), n = m + 2k ; 2. tGENPR(A k m) = O(k n)
A(i) := 1 d := 3. 2. d2 > n, d > n=d],
1.
] n := m + 2k ; 2 6.
i := 1, 2, : : : k ]
3.
p
.
5
3 21 ( . . m = 3 k = 10). d = 3 A(i) = 1, i = 1, : : :, 10. 2 , 3, r=0 j=4( m = d). 4 A, A(4) := 0, A(7) := 0 A(10) := 0, d, d := 5. , 2 5.
76
i = 10 A(9) = 1
,
,
6. 6 A(10) = 0 i=9 19, . 19, 17, 13, 11, 7, 5 3.
2.3.2., , b ; a = mq
mm>1
-
b ma q( ,b f: : : ;3m + a ;2m + a ;m + a a m + a 2m + a 3m + a : : : g), , , , m b ; a. b ma b a (mod m), b a m b a m. , m . Z m = Z m m. , m. -
a. b. c.
a b (mod m) d m, a b (mod d). a b (mod m) a b (mod n), a b (mod m n]). a c (mod m) b d (mod m), a + b c + d (mod m), a ; b c ; d (mod m) ab cd (mod m). b c d. ( .) ab ac (mod m), (mod m=d), d = (a m). , (a m) = 1, ab ac (mod m) b c (mod m). . a b (mod m) k. djm,.
2.3.7.
dj(a ; b). k0 a b (mod d).
d , a ; b = k0 d
,
a,
a ; b = km km,
,
.
3 4 3 6 (mod 6) 4 6 (mod 2), 3 4 3 9 (mod 5) 4 9 (mod 5).
77
.
, . D ab = ac,
2.3.8.a 6= 0 a, b | b = 0. a 6= 0, , ,.
b = c.
, D| D ab = 0 , a=0 , ab = ac, a 6= 0, , a(b ; c) = 0, D , b = c.
a+bs: Z! Z m 2.3.7 , c2a d2b| ), m : m . .
a b Z m a b . a + b = s(a + b) a b = s(a b)., ( , , .m,
, .. ,
s(c+d) = s(a+b) s(c d) = s(a b). m ,
,
-
x m, 0, 1, 2, : : :, m ; 1, , , 0, 1, 2, : : :, (m ; 1)=2 m. a a = mq + r, 0 r < m. r, rm (a) r(a), m. , , m. 2.3.9. b a (mod m) , rm (a) = rm (b). . , a r(a) a (mod m). , b a (mod m)
78
, m j r(b) ; r(a)]. r(a) ; r(b)
rm (a) rm (b), . . 0 r(b) r(a) < m, m , r(b) = r(a). , r(a). . a 0, 1, : : :, m ; 1Z m
,
a = a + mZ,
, m.
f0, 1, : : :, m ; 1g.a a,Z m
|
( .
m ; a, a2Z . m a
)
ax 1 (mod m), ,
2.3.10.(a m) = 1..
m
x y, , (a m) = ax + my, ax (a m) (mod m). (a m) = 1, , x a m. (a m) > 1, ax 1 (mod m), ax = 1 + km (a m) = 1. ,
x,Z = m
2.3.11. f0, 1, 2, : : :, m ; 1gm|Z m
m>1 m. . ,Z m
m , -
s: Z
.
Z, m
, , Z . , , a, b c Z s(a) s(b) s(c)] = s(a) s(b c) = s a (b c)] = s (a b) c] = s(a b) s(c) = s(a) s(b)] s(c).
79
, Zp . , Z | . m , m = a b, a b < m s(a) s(b) = s(m) = s(0), s(a) 6= s(0) s(b) 6= s(0), , s(a) s(b) . . p = 5. Z = f0, 1, 2, 3, 4g 5 , 1, 2, 3 4 ( | 1, 3, 2 4 ). Z= 8 f0, 1, 2, 3, 4, 5, 6, 7g , , 2 4 = 0 (mod 8). , Z m (m) , | , (m). 2.3.12. | . . , | , . . = ab, 1 < a b < . a 1 b 1 , (a 1)(b 1) = (ab) 1 = 1 = 0, .Z p
,
m
, ,
m|
| . . 20, 30 10 .. 89 89 178 , 34 1 , 2, 1 3 2, .. 3 9, ..
m.
, , 10 ,
.+ a
9. abc = 100a+10b+c = 99a+9b+(a+b+c).] : = 17, = 17, 34 178 16 16 = 8 = 8 16 7 (mod 9), 7.
80
:
89 87 7743 12 3 (mod 9) 21 3 (mod 9). 10% . m a;1 (mod m) = 17 = 15 17 15 = 255, , , , -
7743
9 10k 1 (mod 9), k 0.
.
.. 2.3.10, m a;1, .
-
a, (a m) = 1 , my , , m. , ,x| , ,
ax + my = 1 , ax = 1 ; my ax 1 (mod m) L(b) = 1 , 1.] a
.
GF(11)
4 11 4, q a0 a1 x0 x1 y0
4;1 (mod 11) | 11. y1
:
0 1 2 3
; 11 42 1 3 4 3 1
1 0 0 1 3 0 1 1 ;2 1 1 ;1 ;2 3 0 ;1 4 3 ;11
81
4;1 (mod 11) = 3. , , y0 , x0 , ( . m
3
, 1 = 11 (;1) + 4 3, x0 y0 x. ( 4
;1,
4 11.) -
4). (1640). m| a| , m, am;1 1 (mod m). . a, 2a, 3a, : : :, ma. m . , (i ; j)a m, m a, m i ; j, , i j 1, 2, 3, : : :, m.] m a, 2a, 3a, : : :, ma 0, 1, 2, : : :, m ; 1 , 0 ma m. 0, am;1 (m ; 1)! (m ; 1)! (mod m). , am;1 (m ; 1)! ; (m ; 1)! 0 (mod m), , (am;1 ; 1)(m ; 1)! m. m (m ; 1)! , m am;1 ; 1. 2.3.13. m| , Z m a;1 = am;2 . , , (m | ). 2.3.14 ( ). (a m) = 1, a (m) 1 (mod m). . . m r1, r2, : : :, r (m) rk a, (a m) = 1. ( , , m .) , , . , a (m) r1r2 r (m)
82
r1r2 2.3.7, a
r (m) (mod m). m,
(m);1 .
2.3.15.k, , k . a
a (m) 1 (mod m). Z m (a m) = 1 , m m ; 2, . , O(m). , 2000 , k= . ki2i : SMa m, S Ma a ,
, a (m) ; 1. 1, , k
a;1 = k -
, , :
,
(m) m.] , k
X
0 i n;1
0 a
S ak ,
1
SMa
. -
4;1 (mod 11), 9 1001. SM4 S S SM4 , SM4 , SSSM4, , , , 4, , , 11 , 49 = (42)2 ]24. , GF(11) 42 = 5, 44 = 52 = 3 , .. 4, , 49 = (44)2 4 = (32 )4 = 9 4 = 3, , 4 11 3.
. GF(11)49,
m.
,
83
.
, .
k , , .
(Exponentiate) P : a, k m a | Z , k = 0 i n;1 ki2i , m k m ; 2, (m) ; 1. ;1 , : a a m, a;1 = ak Z. m 1. ] K := k B := 1 A := a. 2. ] q := K=2] r := K ; 2 q K := q r = 0, 5. 3. m] B := A B (mod m). 4. ?] K = 0, a;1 (mod m) := B. 5. m] A := A2 (mod m) 2. E. k n , j . n+j , 1. j , n, O(2 n) = O(2 log2 m) ( ) = O(log2 m) tE = O L(m)] . , E , Z. m a, b. , 3 B := A + B 5 A := A + A 1 B := 1 B := 0 K := k | K := b , B = a b. , , , 2 .
E.
84
.2 :
. a 38 76 152 304 608
38 19, a b. 2 a b , a . b 19 9 4 2 1 38 76 608
1 1
2 , b,
b.
b am;1 .
722 = 38 19 , a b = (2a) (b=2), , a b = (2a) (b ; 1)=2 + a, b .
.
| m.
1 (mod m)) .
a ,
, ,
am ; a ( am ; a
m,
m
25 ; 2 2m;1 .
, m| , : 22 ; 2 7;2 5, 2
7,,
1 (mod m). ,
? 2, 23 ; 2 3, 2, 3, 5, 7 . : m m 2m ; 2, , 341 ,
|
, . | , 1819 . , 2341 ; 2 341, , 11 13. 210 1 (mod 341)
, 2340 (210)34 134 1 (mod 341) 341 j (2341 ; 2):
85
, . a. 2,
91, , , 341 91 | , , 210
2340 (mod 341), 1 (mod 341). , a, ( . 5). , -
.
a
,
, 391 ; 3
3.
,
( .
, ,
a 2.3.28). 561 = 3 11 17 1729 = 7 13 19, a. ,
,
2.3.16 (m|.
, 3) (m ; 2). 3 5 ,
m| . ). (m ; 1)! ;1 (mod m) . , m 2 3 4 m( . 2 3 4 5, .7 7. 2 4, ,
m=7
(m ; )
1 (mod m), (m ; 3)=2 . 2 3 4 2 3 4 m ; 1,
(m ; 3) (m ; 2) 1 (mod m)
(m ; 3) (m ; 2) (m ; 1) = (m ; 1)! m ; 1 ;1 (mod m): f(m) = sinf$ (m ; 1)! + 1]=mg. , f(m) m| . , -
,
86
1)
, 2.3.11,Z, 8
m.
(m;1)! |
.Z m
,
.
, . 8,Z. m
,Z m
2, 4 6
.
. 2, 4 6 1, 3, 5 7 , 1, 3, 5 72)
, (m) , * 1 3 5 7 , ..
Um = f a: (a m) = 1 g (m). , f 1, 3, 5, 7 g, , , . 2.3.1, . 1 1 3 5 7 3 3 1 7 5 5 5 7 1 3 7 7 5 3 1 U8 . , .. Z, m
U8 -
. 2.3.1.G G.1) 2)
G| a a a (k )
. ak a0 = e |
n
a .
G
. m;2 .|
, ..
Group of units | . .
.|
.
.
87
,
2.3.17..
a
G| G , G.
,
an = e.
n
-
a2 , : : :, an | a an ( ),
,
fa1, a2 : : :, an g.( (
. a1 , a a1, a a2 : : :, fa a1, a a2 : : :, a an g , ). ( . ) -
, (Cnilds, 1979)). G| n , a 2 G S = fk an = e, S S k0 , a. , a, 1, a, a2 : : : . , Um , m. , m, m . , . k0 | . , , ai a3,
1: ak = e g. m. . ,
,
2.3.18..
G| a
G. ,
,
k0 j n.
n
e, a, a2 , : : :, ak0 ;1 k0 G, , a2 . a2 , a2 a, a2 a2 , : : :, a2 ak0 ;1 | k0 k0 G. G , . , G n
88
. k0 j k.
, k0 |
.
n
G a
k0 j,
aj |
2.3.19..
G ak = e,
k = k0 q + r, 0 r < k0. r = 0, , , r > 0. ak = r = e. k0 q+r = (ak0 )q ar = e, , a , , a , k0 | ak0 = e. G = Um , , )| , a U8 , 8: Um (m). . U8 1 a a2 a3 : : : 4 1 1 1 U8 . ,.
(m). , , ..
,
, 2.3.14 ( U8 -
.
n = 1 2 3 3n 3 1 3 5n 5 1 5 7n 7 1 7 2.3.18).
5 3 5 7 2,
6 1 1 1 U8 8.
7 3 5 7
::: ::: ::: ::: 2, ,
(8) = 4 ( , , m a > 0.
2.3.20.
Um 1, 2, 4, pa m.
2pa,
p|
m
(LeVeque, 1977). , 18 = 2 32. 8| , 2.3.20 U18 | U18 (18) = 6
,
.
,
89
,
U18 = f1, 5, 7, 11, 13, 17g, 5 , n = 0 1 2 3 4 5 6 ::: 5n 1 5 7 17 13 11 1 : : :
2.3.21.x = 1. a;1 .)
2.3.20 m| x2 = 1 Um
18. , , Um
11
m. r
.(
a m Um , | U18 5;1 11 (mod 18), , U18 ? gcd k (m)] = 1, )Z . 18
r0 = rk r0 (18) = 6 , 11,
, 2 11. , , gcd k (m)] = d > 1, r0 = rk (m)=d. , , (m)=d r0, . . (r0 ) (m)=d = r (m)k=d 1k=d = 1 (mod m). , . k (m)], (r0 ) (m)k=d = r (m)k=d = r k (m)] = r (m)j 1j = 1 (mod m). , : : : (mod 8), m 6x 4 (mod 8) 6x 5 (mod 8) x 0, 1, : : : 7 (mod 8)]. , ax b (mod m) (a m) j b. m=d, d = (a m)
(m) ( (5 6) = 1, 55 . (m)]. , U18
, k > 1, ,
U = 18
m.
. x 2, x 6, . m ,
2.3.22.d .
,
90
b y, 2.2.4 ax + my = b , (a m) j b. , x ax b (mod m), z x m=d, d = (a m). z = x + w(m=d) w2Z , az = ax + aw(m=d) = ax + mw(a=d) ax b (mod m) , az b (mod m). , ax az b (mod m). ax ; az b ; b 0 (mod m) , m j a(x ; z). 2.2.6 m=d x ; z, , , x z (mod m=d). . 270x 36 (mod 342). , , (;5) 270+4 342 = 18, 18 j 36. 2.3.22 , 19 = 342=18. (;5) 270 + 4 342 = 18 2 = 36=18 (;10) 270 + 8 342 = 36, , (;10) | 342. 19 , 9, 9 ;10 (mod 19). 342 9, 28, 47, 66, 85, 104, 123, 142 . 2.3.22 . ax 1 (mod m) 2.3.23. , (a m) = 1. a;1 (mod m) m a m. . 2x 1 (mod 26) , (2 26) = 2. : x, , 2x ; 1 = k 26, | , | . m. ( . 6). , , m m = pe1 pe2 pek ( 1 2 k (mod m) , ax + my = b..
,
ax
91
),
6 = 2 3. , x 1 2 Z x2 2 Z , 2 3 Z (0 0), (0 1), (0 2), 6 (1 0), (1 1), (1 2). : + , (x1 x2) (y1 y2 ) = (x1 y1 x2 y2 ), x1 y1 Z( 2), x2 y2 | Z ( 2 3 3) , (0 2) (1 2) = (0 1), 2 2 3. . (x1 x2),
2.3.25.) ,Z=Z 6 2
Z m Z ei . pi
(Z, 3
-
2.3.24 ( m2 , : : :, mk | M = m1 m2 mk .
M x x ::: x a1 (mod m1 ) a2 (mod m2 ) ak (mod mk ): :
). > 1,
m1 ,
, , 0 x M ; 1, xi ai (mod mi ), i = 1, 2, : : :, k, Z1 Z2 Z k. m m m ..
(a1 , a2, : : :, ak ),
x,
Z M
x, 0 x M ; 1, x ai (mod mi ), i = 1, 2, : : :, k. 2 . . x a1 (mod m1 ) x x = a1 + m1 q, q . q x x a2 (mod m2 ), x = a1 + m1 q a2 (mod m2 ), q (m1 );1 (a2 ; a1) (mod m2 ). ( , m1 m2 . .2 , MODINV.) , q = m;1 (a2 ; a1)+rm2 r. 1
92
x (mod m1 m2 ), m1 m2 . , .
q r.
x = a1 + m1 q,
, x = a12 +r(m1 m2 ) , x a12
,
x,
x a12 (mod m1 m2 ) , -
, x0, 0 x0 M ; 1, , x0 ai (mod mi ) i. x ; x0 0 (mod mi ) i, , mi j (x ; x0) i. M j (x ; x0) , jx ; x0j < M, x = x0. , 2.3.24 mi , , (mi mj ) j (ai ; aj ) i, j. , m1 , m 2 , : : :, mk ] mi . . 10 .
.
x 1 (mod 2) x 2 (mod 5) x 5 (mod 7): 2.3.24, 1 + 2q q, 2 (mod 5), , , x 2q (2 ; 1) (mod 5). 2 (mod 5), 3 (mod 5) q = 3 + 5r x = 1 + 2q. -
3, , ,q r. , x = 1 + 2(3 + 5r) = 7 + 2 5r, . . x 7 (mod 2 5). x
x 7 (mod 2 5) 5 (mod 7). x = 7 + 2 5q 5 (mod 7), 2 5q (5 ; 7) = ;2 5 (mod 7). 10 7 3 7, 5. q 5 5 (mod 7) 4 (mod 7), q = 4+7r r. , x = 7 + 2 5(4 + 7r), x 47 (mod 2 5 7).
93
, 2.3.14,
47 = 1 + 3 (2) + 4 (2 5), q. m = pe1 pe2 1 2 pek . k
3 4 , .
x xi (mod pi i ), i = 1, 2, : : :, k, (.. (x1, x2 , : : :, xk ), xi 2 Z ei piZ ei , pi
) Z m i = 1, 2, : : :, k. , + , (x1 , x2, : : :, xk ) (y1 , y2 , : : :, yk ) = (x1 y1 , x2 y2 , : : : , xk yk ), i = 1, 2, : : :, k. 2.3.241 i k Z ei . pi
2.3.25. x2Z m e
, (x1, x2, : : :, xk ),
.
. 2.3.25, Um =1 i k Upei . iZ m
=
, ),
2.3.25 (
x. x, 0 < x < M, . mi , M=
. 2.4, m1 m2
mk (mi mj ) = 1
i 6= j,
.
m1 = 3 m2 = 5 6 = (0 1) 7 = (1 2) 0 + 1 (mod 3) 1 + 2 (mod 5)] = (1 3), 13. , (1 3) , , . , .. 1, 2, 3, 5, , .
?
-
6, : : :
>
1
.
,
94
2.3.26.bq
), , b b (mod m). , m = p1p2 pk pi q = p1 ; 1 p2 ; 1 : : : pk ; 1] + 1 , bq b (mod m) b. . , m , .. m = p1p2 pk . , b2Z q>1 bq b (mod m) , bq b (mod pi) i = 1, 2, : : :, k. q = qm +1, qm = p1 ; 1 p2 ; 1 : : : pk ; 1] p1 ; 1 p2 ; 1 : : : pk ; 1. , qm = (pi ; 1)qi i qi bq = bqm +1 = b(pi ;1)qi +1 = b b(pi ;1)qi . (b pi) = 1, bpi ;1 1 (mod pi), b 0 (mod pi ). bq b (mod pi ) i = 1, 2, : : :, k, , , bq b (mod m). , m , ..m= pe1 pe2 pek , ej > 1 j, 1 < j < k. 1 2 k x=b fx 0 (mod pi), i = 1, 2, : : :, k, i 6= j, x pej ;1 (mod pj )g. j , m b, pej b, m j bq q > 1. , bq b (mod m) q > 1. , ( 2.3.24). , , , . : x a (mod m1 ) x b (mod m2 ) (m1 m2 ) = 1: , 2.3.24. MODINV, ( ) m2 m1 ( . .2 ). ,
,
m q>1(
-
95
GCRA2. Chinese Remainder Algorithm-2 congruences)
(Greek-
: a, m1 , b, m2 , , x a (mod m1 ) x b (mod m2 ), m1 , m2 | , , (m1 m2) = 1 m1 , m2 > 1. : x| m1 m2 . 1. a > 0, ] x := MOD(a m1 ). 2. m;1 ] m;1 := MODINV(m1 m2). 3. q] q := MOD m;1 (b ; x) m2 ]. 1 4. ] x := x + m1 q ( 0 q < m2 , x 0 x < m1 m2 ). GCRA2. , 1 2 1, ( ). 3 , 4| . m1 m2 ( , x x ::: x a1 (mod m1 ) a2 (mod m2 ) ), , , tGCRA2 (a m1 b m2) = O L(m1 m2)]: :
x0
ak (mod mk ) (m1 m2 ) = 1 i 6= j: , GCRA2 . , x0 | m1 m2 . m1 m2 m3 x x0 (mod m1 m2 ) x a3 (mod m3 ) . .
96
GCRAk. Remainder Algorithm-k conqruences):
k
(Greek-Chinese
ai , mi , , x ai (mod mi ), i = 1, 2, : : :, k mi , mi > 1 (mi mj ) = 1 i 6= j. : x| m1 m2 mk k . 1. ] m := 1 x := MOD(a1 m1 ). 2. GCRA2] i 1 k;1 fm := m mi m;1 := MODINV(m mi+1 ) q := MOD m;1(ai+1 ; x) mi+1] x := x + mqg. 3. q] q := MOD m;1 (b ; x) m2 ]. 1 4. ] x. GCRAk. , GCRAk , GCRA2. M = m1 mk , iL(m1 m2 mi )L(mi+1 ). P , (k ; 1) , L(M)f 2 i k L(mi )g Q L(M) L( 2 i k mi ) ( , L 2 (M), ) L tGCRAk (ai mi i = 1 2 : : : k) = O L2 (m1 m2 , x 6 (mod 8) x 11 (mod 7), 840 x = 718 718 = 13 + 7 15 + 5 (15 8). , x = q1 + q2 (m1 ) + q3 (m1 m2 ) + qi q1 = MOD(a1 m1 ). 2.3.24. x , qk (m1 m2 mk;1) mi , mk )]: ;2 (mod 15), k ,
(Asmuth, C., Bloom, J.: A modular approach to key safeguarding. Mathematics Department, Texas A and M University, College Station, TX, 77844).
97
,
K| ,
, L L;1
,
k (k > L), ,
. ,
. . fp, d1,
d2, : : :, dk g, , a. p > K. b. d1 < d2 < < dk . c. gcd(p di) = 1 i = 1 2 : : : k. d. gcd(di dj ) = 1 i 6= j. e. d1 d2 dL > p dk;L+2 p dk;L+3 dk. e , L di , p L;1 di . D = d1 d2 dL D=p , L;1 di. r 0 (D=p) ; 1] K 0 D ; 1]. K 0 = K +r p, Ki K 0 (mod di) i = 1 2 : : : k:
. K = 5, L = 2, k = 3, p = 7, d1 = 11, d2 = 13, d3 = 17. , D = d1 d2 = 11 13 = 143 > 119 = 7 17 = p d3 , . r 0 (143=7) ; 1] = 0 19], 2. K 0 = K + r p = 5 + 2 7 = 19:K1 = 19 (mod 11) = 8 K2 = 19 (mod 13) = 6 K3 = 19 (mod 17) = 2: K1 K2 K 0 ; r p = 5. K 0 8 (mod 11) K. K 0 2 (mod 17): , K 0 = 19, , K=
98
2.3.3.. m ., .
: m. m , m m
-
p 2, 3, : : :, b mc. ,.( O(pm). ,
-
m ?
bpmc?), , ..
pm, , , ) .
,
, (
,
m, ,
, :
:
, O(2L(m)=2) . . . -
. (Adleman et al., 1983 Cohen et al., 1982 Dixon, 1984 Lucas, 1961 Pomerance, 1981 Pratt, 1975 Rabin, 1980 Solovay et al., 1977 Williams, 1978), . , . . . m , m j f(m ; 1)! + 1g. 2.3.16 (Wilson). , (m ; 1)! 100 , (m ; 1)! 100102. (m ; 1)! + 1 100102 m. | m
99
, b < m, . . bm;1 1 (mod m) 1 (mod m). b < m, m|
:
m|
,
,
m b m ; 1, . m ; 1, b b .
2.3.27., ..
m| b
m
2.
m , (m) < m ; 1. (b m) = 1, (m) < m ; 1, (b m) > 1, 1 (mod m). , m bm;1 m ; 1. , 1 (mod m) b(m;1)=p p m ; 1. . ). m , m ; 1, m;1 , ( , . b = 2, bm;1 , b,.
m b, (b m) = 1, 1 (mod m) :m ,
b b 1 (mod m). m = 899. -
m ( ,
b < m,
m
, 1 (mod m) , , , b(m;1)=p
(1961). b
m ; 1, 2.3.27). m p , ,
, 899 899 | 899 341,
.
2, 3, 5, 7 11, ,
29 31.
2898 683 (mod 899). , , , . , 2340 1 (mod 341).
100
2 340.
341, 341
,
m ; 1, 35 | ,
340 = 22 5 17, 2340=17 1 (mod 341). 341 | . 40m, , , ..
,
m. m;1
al., 1983) 1982). ,
1980 . ,
,
(Adleman et (Cohen, Lenstra, .
O(L(m)LfL L(m)]g ): LfL L(m)]g , 10999999999. , . , , , : , . (b m), m . , . , , , , LfL L(m)]g = 2, 10999999999, . , . , , -
101
1. 2. 3.km , b. . m|
.
: m| , m| L(m).
-
m ,
k, m b,
, 1 b m. ,
-
b
b, 1 < b < m. b j m, , | , m| , b j m. d(m) | 1 < b < m, p = d(m) ; 2]=m. ,.
.
m
m| . , m . m . , b, m , 341 -
b, 1 < b < m. | , m , m| , m m = pq, p, q | , bm;1 , 1 (mod m), |
(b m) 6= 1, | , ,
m b < m, m ; (m). . , . m| . ,
.
. 2 , .
b
2.
,
102
2.3.28..
m|
n = 2m ; 1
n = 2m ; 1. m = ab, a > 1, b > 1, 2, . . 2m;1 1 (mod m).
2, , -
2,
2ab ; 1 = (2a ; 1)(1 + 2a + + 2a(b;1)): , , 2n;1 1 (mod m), n;1 ; 1. m;1 , , n=2 2 , m j (n ; 1). m 2, m j (2m;1 ; 1) , n ; 1 = 2m ; 1 ; 1 = m;1 ; 1), 2(2 , m j (n ; 1). , , , m j (bm;1 ; 1) b, , (b m) = 1. , bm;1 ; 1 1 m b, (b m) = 1? , m = 561 (561) = 320 b. . , ,| pi , , Q (pi ; 1) j (m ; 1) = ( pi ; 1) pi 561 = 3 11 17. . b, (b m) = 1. bpi ;1 1 (mod pi ). pi ; 1, m ; 1: m;1 bm;1 1 (mod pi) i. , bm;1 1 pi. ( , bm;1 1 (mod pi) x bm;1 ; 1 (mod pi ). , m , x = 0, . . bm;1 1 (mod m).) , b, . | 561 = 3 11 17 : 1105 = 5 13 17, 1729 = 7 13 19 . . . . b m. m ; 1 = t2s , t| xr bt2r (mod m) 0 r < s (xr |
. ,
103
m
x0 = 1, . ,
i, 0 i < s,
, b
m). xi = ;1, | m|
, , m = 561. , 560 = 35 24 r = 4, 3 235 24 = 2560 1 (mod 561) 235 23 = 2280 1 (mod 561) , 35 22 = 2140 67 (mod 561), , r=2 2 , 561 | . , . m| m| (mod m) . bt2i
2.3.29..
,
m| . . r=s , (
. , i, , bt2r;
,
r, 0 r s, i ; 1, ).
1
1 (mod m) 1. ;1 . bt
. , (bt2i 1 )2 =
t
m, , , O L3(m)]. , : m| m| . (Wilf, 1986).
O L(t)] m O L2(m)]. bt2i , i = 1, 2 : : :, r L(m) O L2(m)]. , , , , ,
O L3(m)],
, L(m), r -
, , 1=4
, m,
1=2.
m.
-
104
100 m
m ,
100 , bi , 0 bi m. 1 ; 2;100, .
2.3.4., , . , -
> 1=2. (Dixon, 1981, 1984 Guy, 1976 Knuth, 1969 Lehman, 1974 Morrison et al., 1975 Williams, 1982 1984 Wunderlich, 1979) ( ) , . 2 (1798): u v2 (u ; v)(u + v), (mod m), 0 < u v < m, u 6 v (mod m), m (u ; v), (u + v) (u ; v m), u ; v m, m . u v , . m. n= pm] | pm, , ak = (n + k)2 ; m k( k ). ak (Morrison, pcm = (b b , b , : : :, Brillhart, 1975) Qk : pcm, c0|1 2 bk;1, k ) | , p , Qk k = (Pk + cm)=Qk .] fqi, i = 1 2 : : : j g | , x2 ; m ( . . m qi ). B. ak , , .. ak = (;1)!k0 ak B.
Y
1 i j
qi!ki : Bak -
ek = (wk0 wk1 : : : wkj ) wki !ki (mod 2) i = 0 1 2 : : : j:
105
B2( 0) (mod 2). e0i = 1 2 u= j + 2 BS,
, ,
-
P
),
k:ak 2S ek (0, 0, : : :,
-
Xk:ak 2S
Y
!ki
i = 0 1 ::: j v=
k2S
(n + k) (mod m) ,
Y
1 i j
qiei (mod m)0
u2 v2 (mod m), (u ; v m) m.
1729, . m = 1729, n = 41 ak = (n+k)2 ; m k. a1 = 35, a2 = 120, a3 = 207, a4 = 296, a5 = 387, a6 = 480, a7 = 575, a8 = 672, a9 = 771 . . f2 3 5 7g , 35, 120, 480 672 B, 35 = (;1)0 20 30 51 71, 120 = (;1)0 23 31 51 70, 480 = (;1)0 25 31 51 70 672 = (;1)0 25 31 50 71. 480, S, , 480, (0, 0, 0, 0, 0). e00 = 0, e01 = 8=2 = 4, e02 = 2=2 = 1, e03 = 2=2 = 1, e04 = 2=2 = 1 u = (41 + 1) (41 + 2) (41+ 8) 315 (mod 1729) ( (41 + 1)2 a1 = 35, (41 + 2)2 a2 = 120, (41 + 8)2 a8 = 672 (mod 1729)) v = (;1)0 24 31 51 71 = 1680. u v 1729: u2 = 99225 672, v2 = 2822400 672 (mod 1729). (315 ; 1680 1729) = (;1365 1729) = 91, 91 1729. (Dixon, 1981) , m
.
p O(e( +o(1)) (ln m)(ln ln m) )| L(m). , , o(1) ! 0 , m ! 1. ,
106
, Fm , m 1, 641. . 1729 .
Fm = 22m + 1 1640 . , .
Mm = 2m ; 1, , , ,
F5
2.4
,, . , ( 2.3.25), , , ( )
. ( ) , . . 2.4.1. . , Z
(Gregory et al., 1984 Knuth, 1969 Scott, 1985). e(i1 , i2 , : : :, ih ) i1 , i2 , : : :, ih ,Z ,
, , ( 1=3 = 0:333 : : :), e. ih )Z
, eZ
, -
m, i0j rm (ij ), j = 1, 2, : : :, h. rm (res) . (mod m) ,
e(i1 , i2 , : : :, e(i01 , i02 , : : :, i0h ) Z m ij (mod m), , , i0j em Z m resm , resm res (mod m) = resm res resm res.
,Z Z m
107
Z
? ? y
(e) ; ! ;;; (res) ;;; ;
? ? y. 2.4.1.(e),
(em )Z m
(resm ) x -
. , 7, 20 res,Z. m
..
x ,
,
7.] m,
-
x 7 (mod 13)? . , , x < 13, res, res resm (mod m)
. p| p ,
jresj,, . GF(p)], . :
(Z + ) p ,
,
,
-
.
a (mod p) = a b;1 (mod p)] (mod p) b , b;1 (mod p) | b p, b;1. GF(p) , b a Z . 3=4 (mod 11) 3 4;1 (mod 11) 3 3 (mod 11) 9:
-
108
,
), (3=4) 4 (mod 11) ,
,
(
-
3=4 (mod 11)] 4 (mod 11) 9 4 (mod 11) 3: p , resp = res. resp GF (p), , . . , , -
GF (p), . ,
res. res res p 6= res, , res | GF (p).]
, res
,
GF (p), , ,
1, 2, : : :, (p ; 1)=2g, f0, 1, : : :, p ; 1g . 2.4.2.
,
, Z = f;(p ; 1)=2, : : :, ;2, ;1, 0, p , -
0 1 2 3 4 5 6 7 8 9 10
rrrrrrrrrrr
-5 -4 -3 -2 -1 0 1 2 3 4 5
rrrrrrrrrrr. 2.4.2.p = 11.
,x x=6
17 (mod 11)
x = ;5.
,
109
, ( ,
, . x = 1=3 ; 4=3 ,
) GF (11),
-
.
. x (mod 11) (1=3 + (;4)=3) (mod 11) (1=3 + 7=3) (mod 11) ( (1 3;1 + 7 3;1) (mod 11) (3;1 (1 4 + 7 4) (mod 11) 32 (mod 11) x = ;1. resp = res, . , x = 1=2 ; 2=3, , .
)
3 (mod 11) , GF(p). -
,
res
.,
x (mod 11) (1=2 + (;2)=3) (mod 11) (1=2 + 9=3) (mod 11) (1 2;1 + 9 3;1) (mod 11) (1 6 + 9 4) (mod 11) 42 (mod 11) 9: , : x (mod 11) = (a=b) (mod 11), x = ;2. , b=6| -
110
. a = x (mod 11)] b (mod 11) = 9 6 (mod 11) = 10 (mod 11) = ;1 ( ). , x = (;1)=6, . . , : , m , res resm (m > res). , m , , . . 2.4.3.Z
Z
? ? y
(e) ; ! ;;; (res) ; ! ;;;
Zk m
? ? y. 2.4.3.(e),
(emk )Zk m
(resmk ) -
n
(k = 1 2 : : : n).
e(i1 , i2 , : : :, ih ), i1 , i2 , : : : , ih , emk (i1k , i2k , : : :, ihk ), ijk = rmk (ij ), j = 1, 2, : : :, h k = 1, 2, : : :, n, mk . , emk Z k, Zk m m resmk , k = 1, 2, : : :, n , , res. mk , m1 m2 mn > res. res, , jresj, . . , -
,
111
, 0{29
. ,
,
. , 100, 101, 102, 103
..
-
3, 5, 7. . 2.4.1 3, 5 7. 2.4.10{29 3, 5, 7
N 0 1 2 3 4 5 6 7 8 9
3 0 1 2 0 1 2 0 1 2 0
5 0 1 2 3 4 0 1 2 3 4
7 0 1 2 3 4 5 6 0 1 2
N 10 11 12 13 14 15 16 17 18 19
3 1 2 0 1 2 0 1 2 0 1 , . .
5 0 1 2 3 4 0 1 2 3 4
7 3 4 5 6 0 1 2 3 4 5
N 20 21 22 23 24 25 26 27 28 29
3 2 0 1 2 0 1 2 0 1 2
5 0 1 2 3 4 0 1 2 3 4
7 6 0 1 2 3 4 5 6 0 1 . -
3 5 7 = 105 ,
. 2.4.1 2 1 3]
8
8
= 3 5 7]
8 (mod ) = 8 (mod 3) 8 (mod 5) 8 (mod 7)] = 2 3 1]: , , , . , ,-
112
: : :, mn ], (mi mj ) = 1 , , .
i 6= j, .) n1 n2 , n1
= m1, m2 , = m1 m2 mn . ( ,M = m1, n2 (mod m1 m2 mn ).
2.4.1...
m2 , : : :, mn ]
. = 3 5 7], . 2.4.1, M = 105. 9 114 (mod 105), , , 9 (mod ) = 0 4 2] = 114 (mod ). 2.4.1 , Z = fn (mod ): n 2 Z g M , Z . , M Z ZM , , , M.. ). 2e ; 1, . . u+v u + v (mod 2e ; 1) = fu + v ; 2e g + 1 e ; 1) = (uv mod 2e) + buv=2e c: u v (mod 2 . . .( , u + v ; m, m , ( , u + v < 2e ; 1 u + v 2e ; 1 2e ;1 (.. ) . m= .
u + v m.) , 2e ; 1,
, -
,
113
2(e f ) ; 1.
. 2e ; 1 (mod 2f ; 1) = 2e (mod f ) ; 1. . . 2.4.1 4 = 1, 4, 4] 5 = 2, 0, 5] 1 + 2 (mod 3), 4 + 0 (mod 5), 4 + 5 (mod 7)] = 0, 4, 2] = 9. 1 2 (mod 3), 4 0 (mod 5), 4 5 (mod 7)] = 2, 0, 6] = 20. , . , , M, . , , n, O L(n)] , O L2(n)]. . ,
,
, e f
: (2e ; 1 2f ; 1) = -
, -
. b;1 (mod ), b = b1, b2 , : : :, bn] m = m1 , m2 , : : :, mn ], : , a = a1 a2 : : : an],
b;1 (mod ) = b;1 (mod m1 ) b;1 (mod m2 ) : : : b;1 (mod mn )]: n 1 2 a (mod ) = a b;1 (mod m ) a b;1 (mod m ) 1 1 1 2 2 2 b ;1 (mod mn )]: : : : an bn , , b a, . , . .
, , -
,
114
x
. x (mixed-radix representation). . , x -
, -
x = q1 + q2 m1 + q3 m1 m2 + + qn m1 m2 mn;1 ( ) qi mi qn x. x .( , P x = 1 i n qi Mi , Mi = m1 m2 mi;1 M1 = 1 Mi =Mi;1 i. m1 = m2 = = mn , , m1 = m2 = = mn = 10 .) , 2, , . , . 2.4.2 0, 1, : : : , 5 , qn = 0, 6, 7, : : : , 10 | , qn = 1. , x = a1, a1 , : : :, an ] = m1 , m1 , : : :, mn ], x. x . q1 , q2 , : : :, qn, . , () x q1 (mod m1 ) , , q1 = a1 . . x ; q1 ( q1 , x). x ; q1 = q2 m1 + q3 m1 m2 + + qn m1 m2 mn;1 : ( ) x ; q1 , , ,
115
( ( , x
r,
. , , x ; q1 n ; 1. (m1 );1 (mod r ), ) m1 n ; 1), r = m2 , : : :, 2] ( ) (x ; q1) (m1);1 , q2 . qn 0 1, , .
x = 4, 2, 0, 1] q1 = 4, x0 = x ; 4 = 0, 3, , x0 = 3, 2, 1] r = 5, 3, 2]. q2, (m1 );1 (mod r ) = 7;1 (mod r ) = 3, 1, 1] x0 , 4, 2, 1]. , q2 = 4, , q2 , x00 = 0, 1, 1], x00 = 1, 1]. ( (m2 );1 (mod r ) = 5;1 r = 3 2].) 00 (mod r ) = 2 1], , x , 2 1] q3 = 2. q3 , x000 = 0 1], x000 = 1. ( = 2].) r (m3 );1 (mod r ) = 3;1 (mod r ) = 1], , x000 , 1], , q4 = 1. x , x = ;3. = 7, 5, 3, 2]. 2, 1], , ,
.
2.1.1 1. 2.A, B C a. , 2.1.2. . A (B \ C) = (A B) \ (A C) ( ). B A () B \ A = B. A ; B, A \ B B ; A A B = (A ; B) (A \ B) (B ; A).
b. c.
, ,
116
3. (
n. , n.
.) ,
,
p(n) | ( (i) p(0).
(ii) p(n) . 2.2.1). -
p(n)
4. (
5.
.) C 0 = fx 2 S x 2 C g . = S. , : a. (A B)0 = A0 \ B 0 . b. (A \ B)0 = A0 B 0 . , ,
g
p(n+1), ,
A, B {
S = fm: p(m) m0 .] C S -
,
A B = (A B) n (A \ B). A A A? , (A B) C = A (B C) A, B C. 6. ( .) S { , S jS j. , A B jA B j = jAj + jB j n jA \ B j: jA1 A2 An j.) 7. ( .) . B| B = fS : S | S 2 S g. = , , B 2 B, B 2 B = ( . 1).
(B n A).
A B = (A n B)
-
a. b. c.
2.1.2
1. 2. a. b.
Z Z m =
,
m
2.1.4.
, = f0, 1, : : : M ; 1g.
a = a + mZ.
a -
117
3. a. S = Q x y b. S = Z x y 4. 5. c. S = N x y d. S = N x y,
S( : , , , , ,
),
,
-
5.
jxj y. x;yx y x y . .3 . . ,
, : a. = f(0 0) (1 1) (2 2) (4 4) (6 6) (0 1) (1 2) (2 4), (4 6)g. b. = f(0 1) (1 0) (2 4) (4 2) (4 6) (6 4)g. 6. fa b cg fd eg fa b c d eg. . (b) 5.2 . (a) 4n . log;1? . A B , }(~ ) n -
S = f0 1 2 4 6g.
7. 2.1.3n = f1 : : : ng | ~
3:,
1. 2. 3. 4. 5.
n ,
~= . 0 2.1.9. n!
n 2n , , . .
Bij (~ ) n
log: R+ ! R
S , , f : S ! T.
f :S!S . f(A \ B) S , ,
a. f b. f 6. f| f(A) \ f(B)f
, f(A \ B) = f(A) \ f(B) .
118
7.
f :S !T g:T !U | . a. , g f f. b. , g f . . , g f g. d. , g f . 8. f| f : S ! T. S : xy f(x) = f(y). , .
, , , , f , g
2.2.1
1. 2. 3. 4. 5. 6.
, , , ,
,
7. 8. y) = 1. 9. a1 : : : an | 10.
ajb bja, a = b. d j a b, d j ax + by x, y. a j bc (a b) = 1, a j c. ( . 2.2.3.) , a b , m = ab=(a b) | a b. b 2.2.4. ( . ax + by = 0 x = ;nb=d, y = na=d n.) , gcd(a b) = gcd(a b + ax) x. , d = gcd(a b) d = ax + by, gcd(x, . , gcd(a1 , : : : an) = gcd a1, gcd(a b) = 1 0 a b, ax+by = Fb. N, 0
.
a1 x1 + + anxn = b , gcd(a1 , : : : an) j b, gcd(a2 , : : : an)], n 2. 1, 2.2.3. x, y,
a.
FN , | 1,
. N,
119
,
, 3 F3 = 0=1 1=3 1=2 2=3 1=1: . F1 = 0=1 1=1 ,
.(
a=b
gcd(a b) = 1.) FN N): ( -
b.a=b a0=b0 . (
(a+a0)=(b+b0) a=b
F3, 0/1 1/1:
(a + a0)=(b + b0) 0 =b0.) a ,
0=1 1=2 1=1 ( ) : 0=1 1=3 1=2 2=3 1=1: , FN FN ;1 (a + a0)=N FN ;1 , a0 =b0 N. , F4 F3 , 1/4 3/4 ( F4 = 0=1 1=4 1=3 1=2 2=3 3=4 1=1:
a=b : 2/4?) , ,
c. d.(
, a=b < (a + a0)=(b + b0) < a0 =b0: ( ,.
,
a=b < a0 =b0 ,
a0b ; ab0 = 1:
a=b a0 =b0 | ), ,
e.
7y = 1.
F7
.) 5x +
120
2.2.2f0 = 1, f1 = 1, fn+2 = fn+1 + fn , n 0. , gcd(fn+1 fn )? gcd(fn+1 fn )? 2. ( .) , gcd(fm fn ) = fgcd(m n). ( . m n ( ): fm;1 fn + fm fn+1 = fm+n fn j fkn.) 3. a. , , , ak+2 < (1=2)ak , , . ( k > 1:.
1.
, ak+1 ak .)
. (a) , .. ,
b. 4. 5. EA.
, (a) 34,21 (b) 136,51 (c) 481,325 (d) 8771,3206. ak = ak+1 qk+1 ; ak+2 ak+2. ak+2 < (1=2)ak+1. . d = gcd(a b), a b a > 0, b > 0 gcd(a b) = gcd(jaj jbj).) 1] a = b, 2] a b
: ,
ak = ak+1qk+1 + ak+2 , , . 4, , , d := a , . ( ,
6.
3] 4]
d := gcd(a=2 b).
. d := 2 gcd(a=2 b=2). a, , , ? , a > b,
d := gcd(a ; b b).
121
7.| , ,
, . , .
.
| -
, = . ( , (a + bi)(a ; bi) = a2 + b2 | , (c + di)=(a + bi) (c + di)(a ; bi)=(a2 + b2).) gcd( ) , , . ( , j j| 0, . . .) gcd , 1 i , , gcd. , , . ,. .4
j,
, = , . -
,
;1=2 1/2,,. 3.2.1.)Z
, ,
8. 9. 10.
gcd(12277 399 + 20i). ( fn fn+1 | . gcd 1) = ad ; 1. ,
,
gcd(fn+1 fn ) = 1.
( . gcd(am ; 1 an ; 1) ar ; 1 | , m ; 1 an ; 1, a , m > n. r m n?) (2) P (4) 11. , H1 = d>1 1=d2 = $2 =6, H1 = $4 =90, (8) (6) , H1 = $6 =945 H1 = $8 =9450, , , ( ) . ( . .1
. d = gcd(m n) a > 1,
gcd(am ; 1 an ; , gcd(m n). ,
122
. 2.2.2P : Stark E.I., The Series d>1 k;s , s = 2 3 4 : ::, Once More, Mathematics Magazine 47, 1974, 197{202.)
2.2.3
1. 2. 3. 4. 5.
XEAd = ax+by, d = gcd(a b)? d gcd(a b)?
a. 1 = 3x + 5y. b. 1 = 12x + 21y.
217, 413. x, y, , a, b, x, y, d | . 2a 2b. : (a) 3x + .6 d 12277, 399 + 20i. . 2.2.2, ? ( a b?) ,
, 2y = 5, (b) 2x + 6y = 7. 6. ,
7. 1. 2. 3. p
.
2.2.4, 5 = (2 4, 4 : : :). = 2 + 1= 4 + ( ; 2)].] ,
2.2.12 2.2.13. 2.2.15. p 5 . , 14/3 3/14 (2 1 4) i i1 , i2 i3 , i2 > i3 .
4. a. 5. 6.
b.
. (0 1 1 100). , (i i2 ) < (i i3) i1 i2 : : : in j | ,
(i i1 i2) > (i i1 i3 ): .
(i0 i1 i2 : : : in) > (i0 i1 i2 : : : in + j)
123
7.
.
, n , , n i0 i1 i2 : : : in j0 j1 : : : jn+1 | (i0 i1 i2 : : : in) > (j0 j1 : : : jn + j)?
.
8. a. b. c. d.(1 (2 (2 (2 1 1 3 2 1 1 1 2 , 1 1 1 2 :::). :::). :::). :::). qn=qn;1 = (cn cn;1 : : : c1)
,
-
9. 10. 11. 12.
n 1.
-
pn=pn;1
, p0=q0, p1 =q1, : : : . ,
: 2, 2 ; 1, 2=2, 3. pk =qk . ck , -
p p
p
p
,
c0 0.
qnjqn;1 ; pn;1j + qn;1jqn ; pn j = 1:
2.3.1 1. , n n < p n! + 1. 2. , p pn n. 3. p1 = 2, p2 = 3, : : : | a. pn+1. , , , 2pn + 1..
p, n
, , -
p,
4.
b. pn+1 p1 + p2 +
, n p < 2n (Niven, Zuckerman, 1980).] + pn n > 1. , . . pp 6= r=s.
,
n>1
-
-
124
5. n (n) 6. (n) = 2, n? 7. , p ; p ;1 = p (1 ; 1=p). 8. (n) n 70 80. 9. , n, (n) 10, , . 10. , n = p q, p q{. (n),Zi].
? (p ) =
n .
, n (n) (n) p q
,
pq p q 4.2.2. n -
11. 12. 13.
. 11{15 ( .
14. 15..
z| ( . . z 2 Zi]) n(z) = p | Z , z| Zi]. f , z = x + y, n(z) = x2 + y2 = (x+iy)(x ; iy)]:g , 1+i, 1+2i 2 ; 3i | Zi]. , z| Zi], p Z , , z jp Zi]. , p| Z , p| Zi], p = u v, u v | Zi]. ( , p Zi], , p . .) , p| Z p 3 (mod 4), p Zi]. ( ) , p| Z p 1 (mod 4), p = x2 +y2 , x, y | , ..p Zi]. ( . .) ,
.7
. 2.2.2.)
1. 2. 3.
2.3.2,
. 5(b)
. 2.3.4.
5
2.3.7. 2.3.8. n, 5
n8 ; 1.
125
4. 5. 7. 6. 7. 8. 9. 10.Z m
247 (mod 23) , , 19. , G|.
n
n7 ; n
.
2, 3, 6
E.]s=(r s). s
, 7, 8 9 ,
m ; 1. ar
(m ; 1)
a|
, (mod m2 )
11. 12. (
27. 2.2.4 , x a (mod m1 ) x b (mod m2 ) , (m1 m2 ) j (b ; a). m1 m2 ], m1 m2 . ( . , x x0 (mod m1 ) x x0 , x x0 (mod m1 m2 ]).) 2.3.25. . 6.) : x 2 (mod 3) x 3 (mod 5) x 2 (mod 7):
13. a. b. 14. 15.,
, , , , 9
. . 3 3,
,
-
, 9. a. m = (p1 )e1 (p2 )e2 : : :(pk )ek . x 1 2 : : : k, (mod m) .( m| ,
, a (mod pei ), i = i x a , S(m) f(x) 0 (mod m), Q S(m) = 1 i k S(pei ):) i
126
b. 16. a.,
,
x a (mod mi ) i = 1 2 ::: k x a (mod m1 m2 : : : mk ]), , : : :] . , x2 1 (mod m)
,
2 p , >0 p>2 x2 1 (mod m) x ; (mod m). , 2k , k 3, (mod m) x 2k;1 1 (mod m). k = 1 k = 2? , , x2 e 1 2 (mod m) , x 1 (mod pi i ) pi , ei > 0 m , , , m r , 2r , m. 2r+(8jm)+(4jn);(2jm) , (kjm) 1, kjm, 0 .)
x
p
x0? (
.
x x0
,
, m , x 1 (mod m) m ,x 1
b.
, (mod m) r
,
, m 2r , .
x2 m
x
3x2 . x (mod 1000)? 17. , x2 ;1 (mod p), p| , , p=2 p 1 (mod 4). ( . , p 6= 2 p 1 4, p 3 (mod 4).) 18. a. , p| gcd(a p) = 1, xn a (mod p) gcd(n p ; 1) ,
c.
127
a(p;1)= gcd(n p;1) (mod p). ( , , , g yn
1 (mod p).
a(p;1)= gcd(n p;1) 6= 1 xn a (mod p) 6= 1 (mod p). a (mod p), b (mod m) 2.3.22).) , , -
a(p;1)= gcd(n p;1) p : , gj ax
,
j (mod p ; 1) .
b.
gcd(a m)
p| x2 a (mod p) a(p;1)=2 ;1 (mod p). (a p
m( (a), gcd(a p) = 1,
-
, , : ,
, a(p;1)=2 1 (mod p) a(p;1)=2 (mod p) . .9 . 2.3.3, p, 1,
sa q
p. q = q1q2 : : :qs , . .a q
;1,qa qj
a|
a| , -
= q,
Q
. =1
a q
= 1, , a
1 j s qj
a
,
|
,
x2 2 (mod 9)
.) 3
; 2 = 1, 9, 12 , 4 -
19. 20.
2 n),
11, 13. bn (mod m)
21.
, 7538 (mod 107). , gcd(a p) = 1, ap;2 a p. , p| . , a;1 (mod p),
(..
E
-
128
(a) a
22.
p. , 12
, , 311 11, 9 .( 2.
p, (b) ( ) , 12 8 11, 13,
a 5 13. .) am;1 1 9 -
23. ( 24. 25. 26. 27.
.) , 1 m (mod m) a(m;1)=p p, , pj(m ; 1), m | .( . , , ak (mod m) 1 k < m.) a. , , a m n k (mod m ; 1), an ak (mod m). b. (a), 7 310000. , , , .. , (m n) = (m) (n), gcd(m n) = 1. ( 2.3.14), , gcd(a m) = 1, an an (mod m) , n n0 (mod (m)). ( . 4.2.2.) a. , ( 2.3.14), m| 2p, p| , ( (m)) a, 1 m, , 1 mod p , | , , p jm. , m = 105, 105 = 3 5 7 (105) = 48 a48 1 (mod 105) a , a12 1 (mod 105), 12 | 3 ; 1, 5 ; 1 7 ; 1. b. (a), 210000 mod 77. , .0
129
28. 29.
3-
4
(
)
5, 7
(
13.
), -
x a1 (mod m1 ) x a2 (mod m2 ) a2 m1 m2 a1 a2 , , m| . p, , > 1, , m ? . . , m| (p ; 1)j(m ; 1) , pjm. m = 561 = 3 11 17 560 a1 -
1.
2.3.3 a. b.( m m :
2. 3. a. b. 4..
3 ; 1, 11 ; 1 17 ; 1.) . 1, , ( b, p, n| b, . . p| b, 2, 5 2, 3 ) 7. ) 7.
15 | 2p ; 1 | b, 2p ; 1.
, n = p(2p ; 1),
1.) 50% , -
m| gcd(b m) = 1.
a.
,
m0 = m=p. b. , 3|
bm ;10
, pjm m 1 (mod p),
m = 3p (p > m = 5p (p >
c.
5|
,
130
d. 5. 6. 7.,
, 5?
3.
91 | 2? p| , , . m , bd 1 (mod m). , , , q = 2p+1? p. ( . . 19 , b (bm ; 1)=(b ; 1) , , b m gcd(b m) = 1, . 2.3.2.) 50% () , b1b2. . b, gcd(b m) = 1, b1 b, p, b () , () , , b . , bp;1 p2 1 (mod p2).
b m = pq | d = gcd(p ; 1 q ; 1). b , d, m ? m , m gcd(b ; 1 m) = 1, , gcd(b ; 1 m) = 1. m| b(m;1)=2 ( . .)
8.
b.
,
9. a.
b.
b2,
,
, m| () gcd(b m) = 1: , b, , b,
;b
(mod m): . 18
() , b,
m
c. d.
b(m;1)=2 6
,
m ,
1 (mod m). m| ,p| b p = ;1 b 1 (mod m=p), . , b
131
10.
m| gcd(b m) = 1 m| b. ( , 91
b| ( ),
b. ,
m b,
,
345 27 (mod 91).) 11. , n| 6n+1, 12n +1 18n+1 | (6n + 1)(12n + 1)(18n + 1) | 3,
, .( .) , , m=
.
12. 13.
14. 15.
16. 17. 18. 19.
, : 1105 = 5 13 17 1729 = 7 13 19 2465 = 5 17 29 2821 = 7 13 31 6601 = 7 23 41 29341 = 13 37 61 172081 = 7 13 31 61 278545 = 5 17 29 113. a. , r rpq (p q | ). b. 3pq (p q | ). c. 5pq (p q | ). , 561 | . 561, . =561 ) a. b 2 (Z Z ( 561 ), 561 | . b. , 561 . , m=p , >1 p| , m b , b. , 65 8 18, 14, 8 18 65. , m 3 (mod 4), m , | b b. , b, , m , b,
132
m. . 7.
,
, ,
. 4.2.2
m = pq RSA.
2.3.4 1. a. b. 2. 3. a. b. 4.. b;1 1? b , n| ),
bm ,
b n bn ; 1 bn;1 + bn;2 + + b2 + b + 1. , bmn ; 1 bm ; 1 . 221 ; b+1 ( . 1 2.).
(
n , bn + 1 n;1 ; bn;2 + + b2 ; b + 1. b , 2n ; 1 | , 2n +1 | n| ), , 2. ( ,
), -
( 232 +1 |
, gcd(b n) = 1 a c| . , ba 1 (mod n) bc 1 (mod n), bd 1 (mod n), d = gcd(a c). 5. . 4, , p| , bn ; 1, (i) pj(bd ; 1) d n, (ii) p 1 (mod n). p>2 n , (ii) p 1 (mod 2n).
6. a. b. 7. 8. 9. 10. a.).
215 ; 1 230 ; 1. 233 ; 1 221 ; 1. 15 ; 1 324 ; 1. 3 512 ; 1. 106 ; 1 108 ; 1.(
. 6{9:
,
, m
m|
, m = pq, p q > 0,
133
m u = (p + q)=2 v = (p ; q)=2 (a), ,
.
m = u2 ; v2 , p= u+v
u, v | q = u ; v:
-
b. c.( ,
k = 1 2 ::: u2 ; n = v 2 u := ak ). (b) 91 323. ak x>1| i. m|.
m ak = pm] + k ,
, ( ,
).
11.
,
,
jp2 ; x2qi2 j < 2x i 12. a. ,2pm. ( b. (a).
pi=qi, , ,
pm
, pi=qi | , p2 (mod m) . 11.) pi , , ( , ,
, pi, . .
, p m .( .) ,
, -
c. 2.4 1.
. 2.2.4). 899 1443. 2.4.1.
134
2. 3. 4. 5.
d| , = 3 5 7], a = 19 b = 23.
xd ; 1 m ; 1 xn ; 1 x = 3 5 7],
m n. . -
a + b, a ; b, a b b;1 (mod ),
= 7 5 3 2] | x = 6 3 1 1]?
(d) ;537.
, : (a) 127 (b) ;127 (c) 537 .
2.2.1 1., b , a = bq + r, 0 r < jbj. QUO, q a r,
a. b. c.
q.
MOD, LAVMOD,;jbj=2 < r jbj=2.
r. r,
,
2.2.2 1. EA, a b0 a] = a 0] = 0
GCDLCM, d = gcd(a b) m = lcm(a b). (0 a) = (a 0) = jaj a.
135
: a 18755 4199 69 1463 b 6727 407 9453 14098 (.. 215 ; 1 = 32767), a=(a b)]b ( . ( gcd(a b), . , 1 500. ,
2.
, 21 35] = 3 35). 500
2.2.3 1.y t,
) 100 (6=$2), . . 0:60793 ( ).
(General Linear Equation)), , a b 0 1 t = 0, ax + by = c x, y |
GLE (, ,
a, b, c, x,
t= , x.
2.2.4. : 168x ; 66y = 42, 426x ; 156y = 128,
ax+by = (a b),
XEA,
-
343x + 407y = 7, 1463x + 4235y = 11. p(x) = cnxn + , + c0, cn > 0 | -
2.2.4 1(, 1798).
136
> 1. (pq) ( (
,
( partial quotiens) ,
) . 7):
-
1. ] (pq) := 1. 2. x := 1+x] i = 0, 1, : : :, n ; 1 ( j = n;1, : : :, i ( ), . 3.1.2 cn +cn;1 + 2. pq,
3.
pq := pq + 1 4. x := 1=x]
.] ?]
) cj = cj +1 +cj . { , pti(1 + x) pti(x), pti(x) p(x) -
+c0 < 0, (cn , cn;1, : : :, c0 ) 1. p(x) = , 100.
, (;c0 , ;c1 , : : :, ;cn) pti(1=1) (reciprocals) pti(x).] . , .
x3 ; 2
2.3.120
1. 2. 2.3.2 1.t, t= ,
< 215,
-
LCE (Linear Congruence Equation), a, b m > 1 x, n 0 1 ax b (mod m) . 2.3.22. ,
,
t = 0, x | n = m=(a m)
137
GLE (36x ;20 (mod 16), 57x 148 (mod 38),
. 2.2.3). : 22x 253 (mod 143), 35x 9973 (mod 12).
2.
LCE, MODINV, a 6= 0 m > 1 x t, 0 ax 1 (mod m) , t= 1 , , t = 0, x | m. :32x 1 (mod 35), 119x 1 (mod 12), 74x 1 (mod 128), 486x 1 (mod 1033). : (iii) x 6 (mod 23) x ;50 (mod 12) x 27 (mod 31): , LAVGCRA, . , p , , n>1 n n n, n. ( .
3. a.
GRAk
(i) (ii) x ;28 (mod 15) x 24 (mod 13) x 18 (mod 22) x 52 (mod 17) x ;1 (mod 19)
b. 2.3.4. p pn n. 2, 3, : : :, .2 . 2.3.1). e ,
,
1.
p, e
p, q = n=pep n.
138
(n
,
, pn < q < n q = 1 q|
FACT,n
> n).
p
-
29957.
215p 1 = 32767. ; < b 215c = 181. 9583, 9973, 16384, 17017
1. . .. , y 2 y, = y ,
, , , , . . , . ,
, y = fx : x 2 xg, = , , y. , . , , , , 1200 ., , 1896 ., . ,
y 2 y? y 2 y, y 2 y = , y 2 y, y 2 y. = . | : , ? . . , . . ,
y 2 y, ,
, 2.
3.
139
, .. 4. 5. 1909 . 6. , (= ( 2, 3, 2 , , 18
, . 1640 . 1949 . (
, , ). .
(R.D. Carmichael), 1)
{ { -
)
)
. 3, 5, 7 , .
,
, (Nicomachi Geraseni Pythagorei Introductionis Arithmeticae, Libri II, Lipsiae, 1866). , .( , .) . . . ., ,. .
. .
.
,
. 19, 756{760, 1979.
, -
, 1988. Adleman L.M., Pomerance C., Rumely R.S. On distinguishing prime numbers from composite numbers, Annals of Mathematics 117, 173{ 206, 1983. Bradley G.H. Algorithm and bound for the greatest common divisor of n integers, Communications of the ACM 13, 433{436, 1970.1)
. . , 34{37,
| III . . . |
.
.
140
Childs L. A concrete introduction to higher algebra. Springer-Verlag, New York, 1979. Cohen H., Lenstra H.W. Jr. Primality testing and Jacobi sums. Report 8218, Mathematical Institut of the University of Amsterdam, Amsterdam, 1982. Dickson L.E. History of the theory of numbers. Chelsea, New York, 1952. Dirichlet P.G.L. Abhandlungen der koeniglichen preussischen Akademie der Wissenschaften, 1849, 69{83. Dixon J.D. Asymptotically fast factorization of integers. Mathematics of Computation 36, 255{260, 1981. Dixon J.D. Factorization and primality tests. American Mathematical Monthly 91, 333{352, 1984. Dudley U. Formulas for primes. Mathematics Magazine 56, 17{22, 1983. Erdoes P. On a new method in elementary number theory which leads to an elementary proof of the prime number theorem. Proceedings of the National Academy of Sciences (USA) 35, 374{384, 1949. Goldstein L.J. A history of the prime number theorem. American Mathematical Monthly 80, 599{614, 1973. Gregory R.T., Krishnamurthy E.V. Methods and Applications of Error-free Computation. Springer-Verlag, New York, 1984. : ., . . . | .: , 1988.] Guy R.K. How to factor a number. Congressus Numerantium XVI, (Winnipeg, 1976) pp. 49{89. Knuth D. The art of computer programming. Vol. 2. Seminumerical algorithms. Addison-Wesley, Reading, MA, 1969. : . . . 2. | .: , 1977.] Lagrange J.L. Traite de la resolution des equations numeriques. Paris, 1798. Lame Gabriel. Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers. Comptes Rendues de l'Academie des Sciences (Paris) 19, 867{870, 1844. Lang S., Trotter H. Continued fractions for some algebraic numbers. Journal fuer die reine und angewandte Mathematik 255, 112{134, 1972. Legendre A.M. Theorie des nombres. Paris, 1798. Lehman R.S. Factoring large integers. Mathematics of Computation 28, 637{646, 1974.
Proceedings of the Fifth Manitoba Conference on Numerical Mathematics
141
LeVeque W.J. Fundamentals of number theory. Addison-Wesley, Reading, MA, 1977. Levinson N. A motivated account of an elementary proof of the prime number theorem. American Mathematical Monthly 76, 225{245, 1969. Lipson J.D. Elements of algebra and algebraic computing. Addison-Wesley, Reading, MA, 1981. Lucas E. Theorie des nombres. Blanchard, Paris, 1961. Mills W.H. A prime representing function. Bulletin of the American Mathematical Society 53, 604, 1947. Morrison M.A., Brillhart J. A method of factoring and the factorization of F7 . Mathematics of Computaion 29, 183{205, 1975. Motzkin T.S. The Euclidean algorithm. Bulletin of the American Mathematical Society 55, 1142{1146, 1949. Niven I., Zuckerman H.S. An introduction to the theory of numbers, 4th ed. Wiley, New York, 1980. Olds C.D. Continued fractions. Random House, New York, 1963. Pomerance C. Recent developments in primality testing. The Mathematics Intelligencer 3, 97{105, 1981. Pratt V.R. Every prime has a succinct certi cate. SIAM Journal of Computing 4, 214{220, 1975. Rabin M.O. Probabilistic algorithm for testing primality. Journal of Number Theory 12, 128{138, 1980. Richards I. Continued fractions without tears. Mathematics Magazine 54, 163{171, 1981. Schroeder M.R. Number theory in science and communication. SpringerVerlag, New York, 1984 and 1986 (2nd ed.) Scott N.R. Computer number systems and arithmetic. Prentice-Hall, Englewood Cli s, N.J., 1985. Sims C.C. Abstract algebra, a computational approach. Wiley, New York, 1984. Solovay R., Strassen V. A fast Monte Carlo test for primality. SIAM Journal of Computing 6, 1977, 84{85 erratum ibid., 7, 118, 1978. Wilf H.S. Algorithms and complexity. Prentice-Hall, Englewood Cli s, N.J., 1986. Williams H.C. Primality testing on a computer. Ars Combinatoria 5, 127{185, 1978. Williams H.C. The in uence of computers in the development of number theory. Computers and Mathematics with Applications 8, 75{93, 1982.
142
Williams H.C. Factoring on a computer. The Mathematics Intelligencer 6, 29{36, 1984. Wunderlich M. A running time analysis of Brillhart's continued fraction factoring method. Number theory Cardondale 1979, Lecture notes in Mathematics no. 751. Springer-Verlag, Berlin, 1979, pp. 328{342.
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