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

Abstract

Let k be a positive integer, and let a, b be coprime positive integers with 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, we prove that if k > 1 and a, b > 2 are both prime powers, then the equation (ak)^x + (bk)^y = ((a + b)k)^z has only one positive integer solution: namely, (x, y, z) = (1, 1, 1). This proves some cases of a conjecture of Yuan and Han.