On a conjecture on exponential Diophantine equations

TL;DR

This paper investigates solutions to the exponential Diophantine equation $a^x+b^y=c^z$ under specific modular and gcd conditions, proving uniqueness of certain solutions and deriving inequalities for potential additional solutions.

Contribution

It establishes the uniqueness of solutions of the form $(2,2,r)$ with odd $r$, and proves uniqueness when any of $a$, $b$, or $c$ is a prime power, advancing understanding of this conjecture.

Findings

Uniqueness of solutions $(2,2,r)$ with odd $r$ under given conditions.

Proved uniqueness when $a$, $b$, or $c$ is a prime power.

Derived inequalities constraining potential second solutions.

Abstract

We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this is the only solution in integers greater than 1, with the possible exception of finitely many values $(c,r)$. We also prove the uniqueness of such a solution if any of $a$, $b$, $c$ is a prime power. In a different vein, we obtain various inequalities that must be satisfied by the components of a putative second solution.