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3 = 0? , , ( )?, 1, 8, 15, 22, 29?, 1 7, 1, 8, 15, 22, , 1, 8, 15, 22, , 7,, ( 2, , 15 ) 1?,, 4, 100, 400 (365 ) 2004, 2000, 1
4 , 7,,,, 31, , 7 ( ) 0, 1, 2, 3, 4, 5, 6 7, 2 7 7, 7 ( 7 ), , = , 2, , 2, 3,, 0 2
5 , 7, 7, 7, = 0 (1 7 0) 7,, n 1 (, n 2), }{{} n = 0 (1 n 0), 1 n 0?, }{{} n = n (1 n n), 0, 1, 2, 1, 2,, 1 n 0,?, ( ),,, 1 n }{{} n = 0 3
6 1 Z 5 2 Z n ( 1 n 0 ) 7 21 Z n 7 22 Z n Z n 1 n Z n 14 3 Z n p = a 2 + b 2 p (1)(2) 22 4
7 1 Z, 1 n 0,,, 0, 1, 2, 3, 1, 2, 3,,,,, Z 3 Z = {, 3, 2, 1, 0, 1, 2, 3, },,,, Z (, a, b, c, ) 11 ( Z ) (I) ( ) a + b ( ) a b (II) (1) a + b = b + a ( ) (2) (a + b) + c = a + (b + c) ( ) (3) a, a + 0 = a (4) a, b a + b = 0 (b a a ) (III) (1) ab = ba ( ) (2) (ab)c = a(bc) ( ) (3) a(b + c) = ab + ac ( ) (4) a, a 1 = a a b a b, a b ab,,,,, 3 Z Zahl 5
8 ,,, 4 12, 2 5 1,, (11) (I) Z, 2, Z, 2, ( 1 2 = 1 ), Z 2, (11)?, Z ( ), a b, a b = a + ( b),, ( (11) II(4)), (11), Z, 2, (11), Z, 2 ( ), a, b, c, (11) (II)(III), (, (II)(3)(4) (III)(4) 0 1,, ), (11) 6, Z (11) 1 n 0, 1 n 0, (11), 1 n 0 Z (11) 4, (11), (1) ( a) = a (2) 0 a = 0 (3) ( 1) ( 1) = 1 (4) ( a) ( b) = ab 5,, 3 Z 6,,, (11), (ring) 6
9 2 Z n ( 1 n 0 ) n (, n 2) 1 n 0, Z n, Z n, n 2, n Z n, n ( 2), Z n n = Z n 2 Z 2 ( ) 3 ( ) Z Z 100 ( ), Z n (1) Z n (2) Z n ( ) (3) Z n (11), 1 n 0 21 Z n Z n, Z, Z, Z ( ), 0, 1, 2,, 1, 2, ( ) ( ), Z n 7, Z n, 3, 2, 1, 0, 1, 2, 3,,,,, 2 ( ), a a Z n, 7 n, Z n,, Z n, n, n, a 7
10 , Z n a (a ), 5 19 ( 5 19 ), Z n 5 19, = 19, Z n, Z n 2 a b, a = b,, 21 (Z n ) a b n, a = b a b n, a b n, 22 (Z n ) a b n, a = b 23 Z n (22), = 14, n n = 2, 14 2 Z 2, 5 = 19, n = 3, 14 3 Z 3, 5 19, 2 Z n, n, , 7, 14 3, 5 19 { Z2, Z 7 Z 14, 5 = 19 Z n (n 2, 7, 14),
11 24 n = 3 Z 3, 6 ( 6) = 12 3, 6 = 6, 8 13 = 5 3, 8 13, 3, 0 1 2, Z 3,, 100,, 3, 2, 1, 0, 1, 2, 3,, 100,, 0, 1, 2, = 6 = 3 = 0 = 3 = 6 = (3 0) = 5 = 2 = 1 = 4 = 7 = (3 1) = 4 = 1 = 2 = 5 = 8 = (3 2), Z 3 = {0, 1, 2} (Z 3 3 ) 25 n = 2 Z 2,, Z 2, = 4 = 2 = 0 = 2 = 4 = (2 0, ) = 3 = 1 = 1 = 3 = 5 = (2 1, ), Z 2 = {0, 1} (Z 2 2 ) (24) (25), Z n, n 0, 1,, n 1 Z n n Z n = {0, 1, 2,, n 1} 26 Z n, n, 1, Z n n ( ) 0 : n 0 1 : n 1 2 : n 2 n 1 : n n 1 9
12 ,, Z n 2 n + 1 (n 2),, n 2 (, 112, 111, 0 ( ), 111, 222, 2 ) n + 1 a 1, a 2,, a n, a n+1 ( ), Z n, a 1, a 2,, a n, a n+1, Z n n, n + 1 Z n, 2 2 a i a j, Z n, a i = a j, a i a j n, a i a j 2 a i a j, n 22 Z n, 2 1, Z n,, 8, 27 (Z n ) a b = a + b a b = a b ( + ), a b = r,, r a b a + b n a b = s,, s a b a b n,,, 8 Z n, +,,, + ( ) 10
13 , a = c b = d } = { a b = c d a b = c d, 28 Z 3, 5 = 8, 2 = 13, 5 2 = 5 + ( 2) = = = 21,, 3 = 21, 3 21 = 18 3, 3 = 21, 5 2 = 8 13, 5 2 = 5 ( 2) = = 8 13 = 104, ( 10) 104 = 114 3, 10 = 104, 5 2 = 8 13, 29 Z n a, b, c, d, a = c, b = d, a b = c d, a b = c d a = c, a c n, a c = np (p ), b = d, b d = nq (q ), a = c + np, b = d + nq 11
14 , a + b = (c + np) + (d + nq) = (c + d) + n(p + q), (a + b) (c + d) = n(p + q), (a + b) (c + d) n, a + b = c + d, a b = c d, ab = (c + np)(d + nq) = cd + n(cq + dp + npq) ab cd = n(cq + dp + npq), ab cd n, ab = cd, a b = c d 23 Z n 1 n 0, Z n (11), 1 n 0 Z n, (27), Z n (11), (11) (+, ) a, b, c, Z n (, ) Z n a, b, c Z n,, 210 Z n (11), (11) (I), Z n (27) (29), (11) (II)(1)(2) (III)(1)(2)(3) Z, Z n, + ( (27) 2 ), Z n (II)(1)(2) (III)(1)(2)(3) (?) 1, (III)(3) (, ) a (b c) = a (b + c) (Z n ) = a(b + c) (Z n ) = ab + ac (Z (III)(3) ) = ab ac (Z n ) = (a b) (a c) (Z n ), a (b c) = (a b) (a c) 12
15 , (II)(3)(4) (III)(4), a 0 = a + 0 = a a a = a + ( a) = 0 a 1 = a 1 = a, Z 0, 1, a, Z n 0, 1, a, Z n (11), Z 0 1, (11) (II)(3) (III)(4),, a, a + x = a x a, a y = a y, 1 x y, 9, Z 0, 1, Z n 0, 1 Z Z n , 1 n 0,,, n, Z, }{{} n = n, Z 1 n n, Z n }{{} n = }{{} n = n = 0, Z n, 1 n }{{} n = 0 9,
16 24 Z n 2, + a b c a b x c, ( + ), 1 ( ) 1 ( ),, 1 1, 1 1,, 3 b 4 c x, b + c = x,,, n, Z n 211, Z n a, a, Z n, 1 1, n, n = 2, 3, 4 (1) n = Z 2 (2) n = Z 3 14
17 (3) n = Z 4 212,, 0 ( ) a, a b = 1 ( ) b (, a 1), a Z 1 (,, 2 2 ) (211), Z 2 Z 3, 0, Z 2 Z 3, Z 4, 1 3 (1 1 = 1, 3 3 = 1), 2, Z 4, Z n?, 213 n ( 2) 10, Z n (field), n, Z n,,, 214, n 2, 1 n 0 Z n, n = 1, 1 = 0?, Z n, n = 1, Z 1, n, 1 (n ),, 20, 2, 3, 5, 7, 11, 13, 17,
18 Z n,, Z n, + Z n n ( 2), Z n = {0, 1, 2,, n 1} a + b = a + b a b = a b, Z, (11), }{{} n = 0 Z Z n, Z Z n a n r a = r a n a = 0 a b n a = b Z n, n 16
19 3 Z n, Z n, Z n 1, a Z n, k, (27) Z n, Z n a a a }{{} k = a } a {{ a } k,,, Z n Z, a k a k, a k a k a k = a } a {{ a } k a k a k, a k = a k,, 31, Z 7, Z 7, 3 ( ) = 10 3, Z 7 10 = 3, 1000 = 10 3 = 10 3 = 3 3 = 3 3 = 27, Z 7 27 = 6, 1000 = 6, 6, Z 7 27 = 1, 1000 = 1, 1, Z n 17
20 4 (1) (2) (1) Z 8 9 = 1, = = = = 1, 1 (2) Z 7 9 = 2, = = = 2 134, 134 = , Z 7 8 = = = (2 3 ) = (2 3 ) = = = = 4, 4 5 n, n 3 + 2n 3 Z 3 2 = 1, n = n 2 1, n 3 + 2n = n(n 2 + 2) = n(n 2 1) = n(n 1)(n + 1) = n 1 n n + 1, n 1, n, n + 1 3, 1 3, Z 3 n 1, n, n , n 3 + 2n = n 1 n n + 1 = 0, n 3 + 2n 3 6 a, b a 2 + b 2 = 2003 a b, Z 4 a b, 0, 1, 2, 3, a 2 b = 0, 1 2 = 1, 2 2 = 4 = 0, 3 2 = 9 = 1 11, = ? 18
21 , 0 1, a 2 + b 2 (= a 2 + b 2 ), 0 (= 0 + 0), 1 (= = 1 + 0), 2 (= 1 + 1) 3, Z 4, 2003 = 3, Z 4, a 2 + b 2 = 2003, a, b, a 2 + b 2 = 2003,, a, b 2003 (?), 31 p ( 2) p = a 2 + b 2 (a, b ) Z 4, p = 1, p 4 1 ( A ) Z 9, 32 a 9,,, a = 8512, 8512 = , a, a = a 0 + a a a n 10 n, a 0, a 1, a 2,, a n 0 9, Z 9, 10 = 10 2 = = 10 n = 1, a = a 0 + a a a n 10 n = a 0 + a a a n 10 n = a 0 + a 1 + a a n = a 0 + a 1 + a a n, a = 0, a 0 + a 1 + a a n = 0, a 9 a 0 + a 1 + a a n 9, a = 37521, = 18 9, , a = , = 31 9, ( ) Z 11, a 11 a 0 a 1 + a 2 a
22 32?,,,, 12,, ( ), 1, Z n, Z n 1, ISBN 1 ( ), ISBN C3041 Y3600E, C3041 Y3600E, ISBN ISBN, International Standard Book Number ( ),, 2 ISBN, ISBN,, (coding theory),, CD,, (cryptography theory),,, 20
23 3,,,, 4, 84, 1, 10 13, 3 ( ), ISBN Z n, ISBN Z 11 14, ISBN 10 a 1, a 2,, a 10, ISBN a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10, 2 1 a 1 a a , Z 11, a 1 + 2a 2 + 3a a 10 = 0 (, a 1 + 2a 2 + 3a a ), a 10, a 1 a 9 1, ISBN, 9 ISBN, (1) ISBN 10 1, (2) ISBN 10 2, 1 (ISBN ),,, 3 3 2, ,,, Z 11, ( B ) 13,, ISBN 14 Z 11, 11 10, 213, Z 11 15, a 10 10, ISBN X ( 10) 21
24 4 41 p = a 2 + b 2 p, 1000, p = a 2 + b 2 (a, b ) p ( p, p = a 2 + b 2?) (1)(2), ISBN a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 2, Z 11, a 1 + 2a 2 + 3a a 10 = 0 ( ) i (1 i 10) a i, a i, 2, Z 11, a 1 + 2a ia i a 10 ( a i = a 1 + 2a ia i a 10 + i(a i a i) = i(a i a i) ( ( ) ) 0 ( i 0, a i a i 0 ), 2, i a i j a j (1 i < j 10),, a i = a j,, a i a j, Z 11, a 1 + 2a ia j + + ja i a 10 ( a i a j ) = {(a 1 + 2a ia i + + ja j a 10 ) (ia i + ja j )} + {ia j + ja i } = (ia i + ja j ) + (ia j + ja i ) ( ( ) ) = (j i)(a i a j ) 0 ( j i 0, a i a j 0 ),, 2 22
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