三项纯指数Diophantine方程的一点注记

A Note on the Ternary Pure Exponential Diophantine Equations

设a,b,c,l是适合a+b2l-1=c2,2|/bc,c≡-1(mod b2l)的正整数.运用初等数论方法讨论了方程ax+by=cz的正整数解(x,y,z),证明了当b≡5或11(mod 24)时,该方程仅有正整数解(x,y,z)=(1,2l-1,2).

Let a,b,c,l be positive integers satisfying a＋b^2l-1=c^2,2｜/ bc and c≡-1（mod b^2l）.In this paper,using some elementary number theory methods,the positive integer solutions（x,y,z）of the equation a^x＋b^y=c^z are discussed.It is proved that if b≡5or 11（mod 24）,then the equation has only the positive integer solution（x,y,z）=（1,2l-1,2）.