A multi-Frey approach to Fermat equations of signature $(r,r,p)$

In this paper, we give a complete resolution of the generalized Fermat equations $$x^5 + y^5 = 3 z^n \text{and} x^{13} + y^{13} = 3 z^n,$$ for all integers $n \ge 2$, using the modular method with Frey elliptic curves over totally real fields. The results require a refined application of the multi-Frey technique, which we show to be effective in new ways to reduce the bounds on the exponents $n$. We also give a number of results for the equations $x^5 + y^5 = d z^n$, where $d = 1, 2$, under additional local conditions on the solutions. This includes a result which is reminiscent of the second case of Fermat's Last Theorem, and which uses a new application of level raising at $p$ modulo $p$.