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

Abstract

In this paper, we give a resolution of the generalized Fermat equations $$x^5+y^5=3z^n\text{ and }{x}^{13}+y^{13}=3z^n,$$ 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 $p$.

Post-publication revised version. No modifications in the numbering compared to the published version (Trans. Amer. Math. Soc. 371 (2019), no. 12, 8651--8677). Fixed some typos. Expanded proofs of Proposition 6, Theorem 7, and Lemma 10. Updated bibliography. Fixed some mistake (coming from the code) in the proof of Theorem 2