中文核心期刊
CSCD来源期刊
中国科技核心期刊
RCCSE中国核心学术期刊

重庆交通大学学报(自然科学版) ›› 2008, Vol. 27 ›› Issue (6): 1168-1172.

• • 上一篇    

关于连续正整数Diophantine方程的正整数解

邹兆南   

  1. 重庆交通大学 理学院, 重庆 400074
  • 收稿日期:2008-09-04 修回日期:2008-10-08 出版日期:2008-12-20 发布日期:2016-11-07
  • 作者简介:邹兆南(1939-),男,四川仁寿人,教授,主要从事数论研究。

Positive Integral Solutions of Diophantine Equation of Continuous Positive Integers

ZOU Zhao-nan   

  1. School of Science, Chongqing Jiaotong University, Chongqing 400074, China
  • Received:2008-09-04 Revised:2008-10-08 Online:2008-12-20 Published:2016-11-07

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

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

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