Generalised Fermat equation: a survey of solved cases

TL;DR

This survey reviews the known solved cases of the generalized Fermat equation, highlighting parameter families with finite solutions and providing references to original proofs to aid researchers.

Contribution

It compiles and summarizes the solved instances of GFE, facilitating easier access to proof techniques and preventing redundant efforts.

Findings

Identifies parameter families with known solutions

Provides a comprehensive list of solved cases

Serves as a reference for future research

Abstract

Generalised Fermat equation (GFE) is the equation of the form $ax^p+by^q=cz^r$, where $a,b,c,p,q,r$ are positive integers. If $1/p+1/q+1/r<1$, GFE is known to have at most finitely many primitive integer solutions $(x,y,z)$. A large body of the literature is devoted to finding such solutions explicitly for various six-tuples $(a,b,c,p,q,r)$, as well as for infinite families of such six-tuples. This paper surveys the families of parameters for which GFE has been solved. Although the proofs are not discussed here, collecting these references in one place will make it easier for the readers to find the relevant proof techniques in the original papers. Also, this survey will help the readers to avoid duplicate work by solving the already solved cases.