On the generalized Fermat equation $a^2+3b^6=c^n$

TL;DR

This paper completely classifies primitive solutions to the generalized Fermat equation $a^2+3b^6=c^n$ for all $n \

Contribution

It provides a complete proof that the only primitive solutions are a specific known quadruple, using modularity of Galois representations and Ellenberg's work.

Findings

Only known primitive solution is (47, 2, 7, 4)

Proof combines modularity of Q-curves with techniques for small n

Results extend understanding of generalized Fermat equations

Abstract

In this paper, we prove that the only primitive solutions of the equation $a^2+3b^6=c^n$ for $n\geq 3$ are $(a,b,c,n)=(\pm 47,\pm 2,\pm 7,4)$. Our proof is based on the modularity of Galois representations of $\mathbb Q$-curves and the work of Ellenberg for big values of $n$ and a variety of techniques for small $n$.