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

## TL;DR

This paper resolves specific generalized Fermat equations using advanced modular methods with multi-Frey techniques, significantly reducing bounds on exponents and extending results to equations with additional local conditions.

## Contribution

It introduces a refined multi-Frey approach to effectively solve certain Fermat equations over totally real fields, improving bounds on exponents and applying novel level raising techniques.

## Key findings

- Resolved equations $x^5 + y^5 = 3 z^n$ and $x^{13} + y^{13} = 3 z^n$ for specified $n$.
- Developed a new application of level raising at $p$ modulo $p$.
- Extended results to equations with additional local conditions on solutions.

## Abstract

In this paper, we give a 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 $n \ge 2$, and all integers $n \ge 2$ which are not a multiple of $7$, respectively, 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 $p$.