指数Diophantine方程4~x+b^y=(b+4)~2

THE EXPONENTIAL DIOPHANTINE EQUATION 4~x+ b^y=(b + 4)~2

本文研究了指数Diophantine方程4~x+b^y=(b+4)~2的解.设b>1是给定的正奇数,运用有关指数Diophantine方程的已知结果以及有关Pell方程的Stormer定理的推广,证明了方程4~x+b^y=(b+4)~2仅有正整数解(x,y,z)=(1,1,1).

In this paper,we study the solution of the exponential Diophantine equation4x＋ by=（b ＋ 4）2.Let b be a fixed odd positive integer with b 1,using some known results on exponential Diophantine equations and an extension of the stormer theorem on Pell＇s equations,we prove that the equation 4x＋ by=（b ＋ 4）2has only the positive integer solution（x,y,z） =（1,1,1）.