phat-trien-tu-duy-sang-tao-giai-toan-dai-so-8

["Website: tailieumontoan.com \uf0db x4 + y4 + z4 = 8z4 + 5 (1). X\u00e9t v\u1edbi m\u1ed9t s\u1ed1 nguy\u00ean a b\u1ea5t k\u1ef3 th\u00ec n\u1ebfu a ch\u1eb5n th\u00ec a = 2k (k\uf0ce Z) \uf0de a4 =16k4 \uf0ba 0 (mod 8) ; n\u1ebfu a l\u1ebb th\u00ec a4 = (2k + 1)4 \uf0ba 1 (mod 8) Do \u0111\u00f3 x4 + y4 + z4 \uf0ba 0 ; 1 ; 2 ; 3 (mod 8) . Trong khi \u0111\u00f3 8z4 + 5 \uf0ba 5 (mod 8) m\u00e2u thu\u1eabn v\u1edbi (1). V\u1eady kh\u00f4ng t\u1ed3n t\u1ea1i c\u00e1c b\u1ed9 ba s\u1ed1 nguy\u00ean (x; y; z) th\u1ecfa m\u00e3n \u0111\u1eb3ng th\u1ee9c x4 + y4 = 7z4 + 5. 26.21. Ta c\u00f3 412 = (40 + 1)2 = 402 + 80 + 1 \uf0ba 81 (mod 100) 414 \uf0ba 812 \uf0ba 6561 \uf0ba 61 (mod 100) \uf0de 415 \uf0ba 61. 41 \uf0ba 1 (mod 100) \uf0de 41106 \uf0ba 41. (415)21 \uf0ba 41 (mod 100) M\u1eb7t kh\u00e1c 574 = 10556001 \uf0ba 1 (mod 100) \uf0de 572012 = (574)503 \uf0ba 1 (mod 100) V\u00ec th\u1ebf A \uf0ba 41 + 1(mod 100). Do \u0111\u00f3 hai ch\u1eef s\u1ed1 cu\u1ed1i c\u00f9ng c\u1ee7a s\u1ed1 A = 41106 + 572012 l\u00e0 42 26.22. Do a + 20 21 \uf0de a \uf0ba 1 (mod 3) v\u00e0 a \uf0ba 1 (mod 7) b + 13 21 \uf0de b \uf0ba 2 (mod 3) v\u00e0 b \uf0ba 2 (mod 7) Suy ra A = 4a + 9b + a + b \uf0ba 1 + 0 + 1 + 2 \uf0ba 1 (mod 3) \uf0de A \uf0ba 10 (mod 3) X\u00e9t a = 3k + 1 ; b = 3q + 2 v\u1edbi k, q \uf0ce N ta c\u00f3 4a = 43k+1 = 4. 64k \uf0ba 4 (mod 7) 9b = 93q+2 \uf0ba 23q+2 \uf0ba 4. 8q \uf0ba 4 (mod 7). Do \u0111\u00f3 A = 4a + 9b + a + b \uf0ba 4 + 4 + 1 + 1 \uf0ba 10 (mod 7) \uf0de A \uf0ba 10 (mod 7) A \uf0ba 10 (mod 3) v\u00e0 A \uf0ba 10 (mod 7) m\u00e0 (3; 7) = 1 n\u00ean A \uf0ba 10 (mod 3.7) Hay A \uf0ba 10 (mod 21). V\u1eady s\u1ed1 d\u01a3 trong ph\u00e9p chia A cho 21 l\u00e0 10. 26.23. 23 \uf0ba 1 (mod 7) \uf0de (23)n \uf0ba 1 (mod 7) \uf0de 23n + 1 = 2.(23)n \uf0ba 2 (mod 7). v\u00e0 23n \u2013 1 = 22.(23)n \u2013 1 \uf0ba 4 (mod 7). N\u00ean A \uf0ba 2 + 4 + 1 \uf0ba 0 (mod 7) ngh\u0129a l\u00e0 A 7. M\u00e0 v\u1edbi n \uf0ce N* th\u00ec A > 7. V\u1eady A l\u00e0 h\u1ee3p s\u1ed1. 26.24. \uf022 n \uf0ceN* ta c\u00f3 20124n \uf0ba 0 (mod 2) ; 20134n \uf0ba 1 (mod 2) ; 20144n \uf0ba 0 (mod 2) ; 20154n \uf0ba 1 (mod 2) . Do \u0111\u00f3 A \uf0ba 2 \uf0ba 0 (mod 2). * Ta l\u1ea1i c\u00f3 2012 \uf0ba 0 (mod 4) \uf0de 20124n \uf0ba 0 (mod 4) ; 2014 \uf0ba 2 (mod 4) \uf0de 20142 \uf0ba 22 \uf0ba 0 (mod 4) \uf0de 20144n \uf0ba ( 20142)2n \uf0ba 0 (mod 4) Do 2013 \uf0ba 1 (mod 4) \uf0de 20134n \uf0ba 1 (mod 4) ; Do 2015 \uf0ba \u2013 1 (mod 4) \uf0de 20154n = (\u2013 1)4n \uf0ba 1 (mod 4) V\u1eady A \uf0ba 2 (mod 4) ngh\u0129a l\u00e0 A chia cho 4 d\u01a3 2. Ta c\u00f3 A 2 ; A \uf02f 22 ; 2 l\u00e0 s\u1ed1 nguy\u00ean t\u1ed1. V\u1eady A kh\u00f4ng l\u00e0 s\u1ed1 ch\u00ednh ph\u01a3\u01a1ng \uf022 n \uf0ceN*. 350","Website: tailieumontoan.com 351"]