Abstract: In this paper,we prove that for the congruence equation x²ⁿ+(x+1)²ⁿ+…+(x+h)²ⁿ≡(x+h+1)²ⁿ(mod 7) there are solutions if and only if (1) when n≡0(mod 3),then h≡0,1,7,8,9,15,16,17,23,24,25;31,32,33,39,41,42,47,48(mod 49);(2) when n≡1(mod 3),then h≢3,4(mod 7);(3) when n≡2(mod 3),then h≢4(mod 7).

Key words: module, minimal nonnegative residue, congruence equation, Fermat theorem

摘要： 本文证明了,同余方程x²ⁿ+(x+1)²ⁿ+…+(x+h)²ⁿ≡(x+h+1)²ⁿ(mod 7)有解的充分必要条件是:(1)若n≡0(mod 3),则h≡0,1,7,8,9,15,16,17,23,24,25;31,32,33,39,41,42,47,48(mod 49);(2)若n≡1(mod 3),则h≢3,4(mod 7);(3)若n≡2(mod 3),则h≢4(mod 7)。

关键词: 模, 最小非负剩余, 同余方程, Fermat定理