, 1. x 2 1 = (x 1)(x + 1) x 3 1 = (x 1)(x 2 + x + 1). a 2 b 2 = (a b)(a + b) a 3 b 3 = (a b)(a 2 + ab + b 2 ) 2 2, 2.. x a b b 2. b {( 2 a } b )2 1 =
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1 x n 1 1.,,., = = = = = = = = = = , 3, 5, 7, 11, 13, 17, 19.,, 2,.,.,.,?.,,. 1
2 , 1. x 2 1 = (x 1)(x + 1) x 3 1 = (x 1)(x 2 + x + 1). a 2 b 2 = (a b)(a + b) a 3 b 3 = (a b)(a 2 + ab + b 2 ) 2 2, 2.. x a b b 2. b {( 2 a } b )2 1 = b( a b 1)b(a b + 1) b. 2. {b 2 ( a } b )2 b 2 = (b a b b)(b a b + b) 2,... x 4 1, x 5 1, x 6 1,?,.,. x 4 1, x 5 1, x
3 x 4 1 = (x 1)(x 3 + x 2 + x + 1) x 5 1 = (x 1)(x 4 + x 3 + x 2 + x + 1) x 6 1 = (x 1)(x 5 + x 4 + x 3 + x 2 + x + 1). x n 1 = (x 1)(x n 1 + x n x + 1),., 4. 4 = 2 2 4?, x 4 1. x 4 1 = (x 2 ) 2 1 = (x 2 1)(x 2 + 1) = (x 1)(x + 1)(x 2 + 1), x 3 + x 2 + x + 1 = (x + 1)(x 2 + 1). n,.. x 5 1, 5,. 3
4 x 5 1 = (x 1)(x 4 + x 3 + x 2 + x + 1) x 6 1, 6 = 3 2. x 6 1 = (x 3 ) 2 1 = (x 3 1)(x 3 + 1) = (x 1)(x + 1)(x 2 + x + 1)(x 2 x + 1). x = (x + 1)(x 2 x + 1),. x = x 3 ( 1) 3 = (x ( 1))(x 2 + ( 1)x + ( 1) 2 ) = (x + 1)(x 2 x + 1) 6 = 2 3?,. x 6 1 = (x 2 ) 3 1 = (x 2 1)((x 2 ) 2 + x 2 + 1) = (x 1)(x + 1)(x 4 + x 2 + 1),? x 4 + x = (x 4 + 2x 2 + 1) x 2 = (x 2 + 1) 2 x 2,,., x n 1, n. x n 1, n = 4, 5, 6., x n 1. 4
5 , x n 1., x 8 1, x 12 1, x 24 1.,. n n x n 1 x n x 1 x 1 2 1,2 x 2 1 (x 1)(x + 1) 3 1,3 x 3 1 (x 1)(x 2 + x + 1) 4 x 4 1 (x 1)(x + 1)(x 2 + 1) 5 x 5 1 (x 1)(x 4 + x 3 + x 2 + x + 1) 6 x 6 1 (x 1)(x + 1)(x 2 + x + 1)(x 2 x + 1) 7 x x x x x 16 1 x 18 1 x
6 . n n x n 1 x n x 1 x 1 2 1,2 x 2 1 (x 1)(x + 1) 3 1,3 x 3 1 (x 1)(x 2 + x + 1) 4 1,2,4 x 4 1 (x 1)(x + 1)(x 2 + 1) 5 1,5 x 5 1 (x 1)(x 4 + x 3 + x 2 + x + 1) 6 1,2,3,6 x 6 1 (x 1)(x + 1)(x 2 + x + 1)(x 2 x + 1) 7 1,7 x 7 1 (x 1)(x 6 + x 5 + x 4 + x 3 + x 2 + x + 1) 8 1,2,4,8 x 8 1 (x 1)(x + 1)(x 2 + 1)(x 4 + 1) 9 1,3,9 x 9 1 (x 1)(x 2 + x + 1)(x 6 + x 3 + 1) 12 1,2,3, (x 1)(x + 1)(x 2 + x + 1) x ,6,12 (x 2 + 1)(x 2 x + 1)(x 4 x 2 + 1) 16 1,2,4 (x 1)(x + 1)(x 2 + 1) x , 16 (x 4 + 1)(x 8 + 1) 18 1,2,3,6 (x 1)(x + 1)(x 2 + x + 1)(x 2 x + 1) x ,18 (x 6 + x 3 + 1)(x 6 x 3 + 1) 24 1,2,3,4,6 (x 1)(x + 1)(x 2 + x + 1)(x 2 + 1)(x 2 x + 1) x ,12,24 (x 4 + 1)(x 4 x 2 + 1)(x 8 x 4 + 1) 6
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8 . 2..,,.,,,..,.,. (2),,.,.????..,.,.,,. (3).,,.?,,. (4),..,,,..,.,.. (5)..?
9 ..,,,,. 3. (1)., (2),,.,.,..,.,,. x 1. n.. x 1 n...? x n 1 = (x 1)(x n 1 + x n x + 1), x + 1., n,.. x + 1 n? 9
10 , n 2, 4, 6, 8...,? 2, 4, 6, 8? 2.. n = 2, 4, 6, 8,, 2m. n = 2m,. x n 1 = x 2m 1 = (x 2 ) m 1 = (x 2 1)((x 2 ) m 1 + (x 2 ) m x 2 + 1)... x 2 + x + 1 n? x 2 + x + 1 n 3,6,9,12,18,24., x 2 + x + 1 n 3. n = 15. (4). n = 15. x 15 1 = (x 3 ) 5 1 = (x 3 1)(x 12 + x 9 + x 6 + x 3 + 1) = = (x 1)(x 2 + x + 1)(x 4 + x 3 + x 2 + x + 1)(x 8 x 7 + x 5 x 4 + x 3 x + 1)) n = 15.. n.. 10
11 n? n = 6 1, 2, 3, 6. x 6 1 x 1, x + 1, x 2 + x + 1, x 2 x + 1. n. 4 n 2 x + 1 x n 1.. n 2, x + 1 x n 1., 2 x F 2 (x) = x + 1, 2 d, F d (x).. d F d (x) 1 x-1 2 x+1 3 x 2 + x x x 4 + x 3 + x 2 + x x 2 x x 6 + x 5 + x 4 + x 3 + x 2 + x x x 6 + x x 4 x x 8 x 7 + x 5 x 4 + x 3 x x x 6 x x 8 x
12 x n 1, n. x 1 = F 1 (x) x 2 1 = F 1 (x)f 2 (x) x 3 1 = F 1 (x)f 3 (x) x 4 1 = F 1 (x)f 2 (x)f 4 (x) x 5 1 = F 1 (x)f 5 (x) x 6 1 = F 1 (x)f 2 (x)f 3 (x)f 6 (x), n x n 1,., x 36 1., x x 36 1 = F 1 (x)f 2 (x)f 3 (x)f 4 (x)f 6 (x)f 9 (x)f 12 (x)f 18 (x)f 36 (x) 36 = x 36 1 = (x 18 ) 2 1 = (x 18 1)(x ) 1. x 18 1 = F 1 (x)f 2 (x)f 3 (x)f 6 (x)f 9 (x)f 18 (x) 2. x = (x 6 ) = (x 6 + 1)(x 12 x 6 + 1) x
13 x = (x 2 ) = (x 2 + 1)(x 4 x 2 + 1) = F 4 (x)f 12 (x). x 36 1 = F 1 (x)f 2 (x)f 3 (x)f 4 (x)f 6 (x)f 9 (x)f 12 (x)f 18 (x)(x 12 x 6 + 1) 36 F 36 (x) = x 12 x 6 + 1, n = 36. x n 1. x n 1 = 0 1 n. F n (x).,.,. 13
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