Abstract: In this paper,we prove that(1) for the Congruence Equation x²ⁿ+(x+1)²ⁿ+…+(x+h)²ⁿ≡(x+h+1)²ⁿ (mod 3)there are solutions if and only if h2, 4 (mod 9);(2) for the Congruence Equation x²ⁿ+(x+1)+…+(x+h)²ⁿ≡(x+h+1)²ⁿ (mod 5)there are solutions if and only if(i) if n≡0 (mod 2),then h2, 3,4,8, 9,10,14, 35,16,18,21,22 (mod 25);(ii) if n≡1 (mod 2),then h≠2 (mod 5).

Key words: module, Congruence Equation, least non-negative residue, Fermat's theorem

摘要： 本文证明了:(1)同余方程x²ⁿ+(x+1)²ⁿ+…+(x+h)²ⁿ≡(x+h+1)²ⁿ（mod 3）有解的充分必要条件是:h2,4(mod 9);(2) 同余方程x²ⁿ+(x+1)²ⁿ+…+(x+h)²ⁿ≡(x+h+1)²ⁿ （mod 5）有解的充分必要条件是:(i)若n≡0(mod 3),则h2,3,4,8,9,10,14,15,16,18,21,22(mod 25);(ii).若n≡1(mod 2),则h2(mod 5)

关键词: 模, 同余方程, 最小非负剩余, Fermat定理