On the Ternary Purely Exponential Diophantine Equation $(ak)^x+(bk)^y=((a+b)k)^z$ with Prime Powers $a$ and $b$

Abstract

Let $k$ be a positive integer, and let $a,b$ be coprime positive integers with $\min\{a,b\}>1$. In this paper, using a combination of some elementary number theory techniques with classical results on the Nagell-Ljunggren equation, the Catalan equation and some new properties of the Lucas sequence (\seqnum{A000204} in OEIS), we prove that if $k>1$ and $a,b$ are both prime powers with $\min\{a,b\}>2$, then the equation $(ak)^x+(bk)^y=((a+b)k)^z$ has only one positive integer solution $(x,y,z)=(1,1,1)$. The above result partially proves that Conjecture 1 presented in (Acta Arith. 2018, 184 (1): 37-49) is true.