The exponential Diophantine equation $(3am^2-1)^x+(a(a-3)m^2+1)^y=(am)^z$

Authors: NAI-JUAN DENG, DAN-YAO WU, PING-ZHI YUAN

Abstract: Let $a,\ m$ be positive integers such that $am\not\equiv0\pmod{3}, 2\nmid a$, and $a>3$. We prove that the exponential Diophantine equation $(3am^2-1)^x+(a(a-3)m^2+1)^y=(am)^z$ has only the positive integer solution $(x,y,z)=(1,1,2)$.

Keywords: Diophantine equation, positive integer solution, Fibonacci number

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