不定同余方程x^(2n)+(x+1)^(2n)+…+(x+h)^(2n)≡(x+h+1)^(2n)(mod17)的解(Ⅰ)

The Solutions of Diophantine Congruence Equation x^(2n) + (x + 1)^(2n)+…+ (x+h)^(2n)≡ (x+h + 1)^(2n) (mod17) (I)

本文证明了:同余方程 x^(2n)+(x+1)^(2n)+…+(x+h)^(2n)≡(x+h+1)^(2n)(mod 17)有整数解的充分必要条件是 (1)若n≡1(mod 8),则h3,4,5,6,10,13(mod 17); (2)若n≡2(mod 8),则h3,4,8,9,1O,14(mod 17); (3)若n≡3(mod 8),则h6,10,11,(mod 17)。

In this paper, we have proved that for the diophantinc congruence equationx2x+ (x+1)2n+…+(x+h)2n≡ (x+h+1)2n (mod 17)there arc integer solutions if and only if(1) when n=1 (mod8), then h3, 4, 5, 6, 10, 11, 13 (mod 17);(2) when n≡2 (mod8), then h3, 4, 8, 9, 10, 14 (mod 17);(3) when n≡3 (mod8), then h6, 10, 11 (mod 17).