Abstract: The positive integral solutions of the Diophantine equation of continuous positive integers xⁿ+(x+1)ⁿ+…+(x+h)n=(x+h+1)ⁿ are still unsettled until now.The latest result known at the present can be described as following: ①If 6≤n≤33;②If n>3 and n is an even integer;③If 34≤n≤200,the equation mentioned above has no positive integer solutions.Variety of sieve methods of prime modulus are adopted and it is proved that the Diophantine equation has no positive integer solutions when n is an even integer fitting with the condition 200<n≤250.The current conclusion has been improved and promoted by this result.

Key words: continuous integers, Diophantine equations, prime modulus, sieve method, integral solutions

摘要： 连续正整数Diophantine方程xⁿ+(x+1)ⁿ+…+(x+h)n=(x+h+1)ⁿ的正整数解问题,是一个迄今为止尚未彻底解决的数论难题。目前已知的结果是:①当6≤n≤33时;②当n>3且为奇数时;③当34≤n≤200且为偶数时,该方程均无正整数解。采用多种素数模筛法证明了:当200<n≤250且为偶数时,该方程均无正整数解。这一结果推广和改进了已有的结论。

关键词: 连续整数, Diophantine方程, 素数模, 筛法, 整数解