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

In this paper, we prove that the only primitive solutions of the equation $a^2+3b^6=c^n$ for $n\ge 3$ 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 $n$.