A study on the exponential Diophantine equation $a^x+(a+b)^y=b^z$

Takafumi Miyazaki; Nobuhiro Terai

2019/95/1-2 (2) — DOI: 10.5486/PMD.2019.8283 — pp. 19-37

A study on the exponential Diophantine equation $a^x+(a+b)^y=b^z$

Authors: Takafumi Miyazaki and Nobuhiro Terai

Abstract:

For any given pair $(a,b)$ of relatively prime integers greater than $1$, we study the Diophantine equation in the title. First, we propose a conjecture to describe the positive integer solutions $x,y,z$ of the equation. Second, we confirm our conjecture for certain pairs $(a,b)$.

Keywords: exponential Diophantine equation, Ramanujan—Nagell equation, simultaneous non-Archimedean valuations

Mathematics Subject Classification: 11D61