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

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Maohua Le

Institute of Mathematics

Lingnan Normal College

Zhanjiang, Guangdong 524048

China

Gökhan Soydan

Department of Mathematics

Bursa Uludağ University

16059 Bursa

Türkiye

**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.

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(Concerned with sequence
A000204.)

Received April 25 2023; revised versions received April 28 2023; August 15 2023; August 18 2023.
Published in *Journal of Integer Sequences*,
August 22 2023.

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