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

Le M., SOYDAN G.

Journal of Integer Sequences, vol.26, no.7, 2023 (ESCI) identifier

  • Publication Type: Article / Article
  • Volume: 26 Issue: 7
  • Publication Date: 2023
  • Journal Name: Journal of Integer Sequences
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus, zbMATH
  • Keywords: Catalan equation, elementary number theory method, Lucas sequence, Nagell-Ljunggren equation, prime power base, ternary purely exponential Diophantine equation
  • Bursa Uludag University Affiliated: Yes

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.