本文研究当n＞k≥2且t≥2时方程k/n=1/x1+1/x2+…+1/xt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三1,5041,6301,8821,13861,15121(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.
Existence of Distinct Positive Integer Solutions to a Generalized Form of Erd?s-Straus Conjecture
In this paper,we study the(distinct)positive integer solution of the equation k/n=1/x2+1/x2+…+1/xt 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.
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