Generalised Fermat equation: a survey of solved cases
Ashleigh Ratcliffe, Bogdan Grechuk
TL;DR
This survey consolidates the landscape of solved cases for the generalized Fermat equation $ax^p+by^q=cz^r$ with $|abc|>1$, organized by signature types and parameter regimes. It highlights Beukers’ parametric descriptions in the $1/p+1/q+1/r>1$ regime, elliptic-curve/finite-rank techniques on the boundary $=1$, and Darmon–Granville finiteness for $<1$, while emphasizing the practical compilation of solved instances and references to methods. The work catalogues extensive results across signatures $(4,4,4)$, $(2,4,r)$, $(2,6,r)$, $(4,4,3)$, and the general $(p,p,p)$, $(p,p,r)$, and $(p,q,r)$ families, drawing on modularity, hyperelliptic/elliptic curves, and descent arguments. It provides a structured resource to avoid duplicate effort and to guide future investigations into remaining open cases, including conjectural frontiers tied to the abc conjecture and Beal-type conjectures. Overall, the paper offers a comprehensive reference framework that connects classical Diophantine techniques with modern modular and computational methods, facilitating targeted exploration of unresolved GFE instances.
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.