Generalized Fermat equation: A survey of solved cases

Abstract

Generalized Fermat equation (GFE) is the equation of the form $a{x}^p+b{y}^q=c{z}^r$, 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.