Abstract: In this paper, we have proved that for the diophantinc congruence equation x²ⁿ+(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 h≢3, 4, 5, 6, 10, 11, 13 (mod 17); (2) when n≡2 (mod8), then h≢3, 4, 8, 9, 10, 14 (mod 17); (3) when n≡3 (mod8), then h≢6, 10, 11 (mod 17).

Key words: module, congruence equation, minimal nonnegative residue, Fennat's theorem

摘要： 本文证明了:同余方程x²ⁿ+(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h≢3,4,5,6,10,13(mod 17);(2)若n≡2(mod 8),则h≢3,4,8,9,1O,14(mod 17);(3)若n≡3(mod 8),则h≢6,10,11,(mod 17)。

关键词: 模, 同余方程, 最小非负剩余, Fermat定理