Abstract: In this paper,we prove that when:(1)n=p-1(p>3),2^α|n(α>0),n≢-2(mod2^α+3);(2)n=k(p-1)(p>3),k(1)/(2)(p-1),2^α|n(α>0),r=[(h+1)/(p)],r(p+1)≢0,2(mod2^α+3),to the Diophantine Equation x^α+(x+1)^α+…+(x+h)^α=(x+h+1)^α there are not any positive integer solutions.

Key words: diophantine equation, congruence, residue, module, prime number

摘要： 本文证明了,当(1)ⁿ=P-1(P<3),2^α|n(α>0),n≢-2(mod2^α+3);(2)n=R(P-1)(p>3),R<₂¹(p-1)2^α|n(α>0),r=[(h+1)/P],r(p+1)≢0.2(mod2^α+3)时,不定力程xⁿ+(x+1)ⁿ+…+(x+h)ⁿ=(x+h+1)ⁿ无正整数解。

关键词: 不定方程, 同余式, 剩余, 模, 素数