See Theorem 1.2 of the paper by [Bennett and Skinner] 1, which settles the problem for $p\ge 7$ (take there $C=1$ and $\alpha_0=2$). (Earlier work of Darmon and Granville (using Faltings's theorem) showed that there are only finitely many solutions.)
Lucia
- 43.7k
- 6
- 193
- 219