Multi-Frey $\mathbb{Q}$-curves and the Diophantine equation $a^2+b^6=c^n$

Abstract

We show that the equation $a^2+b^6=c^n$ has no nontrivial positive integer solutions with $\left(a,b\right)=1$ via a combination of techniques based upon the modularity of Galois representations attached to certain $\mathbb{Q}$-curves, corresponding surjectivity results of Ellenberg for these representations, and extensions of multi-Frey curve arguments of Siksek.