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arXiv · 2206.14032

On a conjecture concerning the number of solutions to $a^x+b^y=c^z$

Maohua Le·Reese Scott·Robert Styer
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Abstract

Let $a$, $b$, $c$ be fixed coprime positive integers with $\min\{ a,b,c \} >1$. Let $N(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of the equation $a^x + b^y = c^z$. We show that if $(a,b,c)$ is a triple of distinct primes for which $N(a,b,c)>1$ and $(a,b,c)$ is not one of the six known such triples then, taking $a 10^{14}$. These results support a conjecture of the last two authors.

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Maohua Le, Reese Scott, Robert Styer. 2022-06-28. On a conjecture concerning the number of solutions to $a^x+b^y=c^z$. https://arxiv.org/abs/2206.14032

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