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

Abstract

Let a, m be positive integers such that $amnotequiv0;(mod3),;2nmid a;$, and a>3. We prove that the exponentialDiophantine 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).