广义Erdös-Straus猜想的互异正整数解的存在性

Existence of Distinct Positive Integer Solutions to a Generalized Form of Erd?s-Straus Conjecture

本文研究当n>k≥2且t≥2时方程k/n=1/x_(1)+1/x_(2)+…+1/x_(t)的互异正整数解,证明若方程有正整数解,则至少有一互异正整数解;当k=5,t=3时,除了n≡1,5041,6301,8821,13861,15121(mod 16380)外方程有一互异正整数解;当n≥3,t=4时,除了n≡1,81901(mod 163800)外方程有一互异正整数解;并进一步指出对于任意的n(>k),当t≥k≥2时,方程至少有一互异正整数解.

In this paper,we study the(distinct)positive integer solution of the equation k/n=1/x_(1)+1/x_(2)+…+1/x_(t)with n>k≥2 and t≥2.We show that the above equation has at least one distinct positive integer solution if it has a positive integer solution.When k=5,we show the above equation has at least one distinct positive integer solution for all n≥3 except possibly when n≡1,5041,6301,8821,13861,15121(mod 16380)with t=3,and for all n≥3 except possibly when n≡1,81901(mod 163800)with t=4.Furthermore,we point out that the above equation has at least one distinct positive integer solution for all n(>k)when t≥k≥2.