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

TL;DR

This paper proves that a specific exponential Diophantine equation involving prime power coefficients has only one positive integer solution under certain conditions, supporting a related conjecture.

Contribution

It establishes the uniqueness of solutions for the equation when coefficients are prime powers greater than 2, using elementary number theory and properties of Lucas sequences.

Findings

The equation has only the solution (1,1,1) under given conditions.

Supports the conjecture proposed in previous research.

Uses a combination of classical and new number theory techniques.

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.