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

Nicolas Billerey; Imin Chen; Luis Dieulefait; Nuno Freitas

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

Number Theory 2024-07-09 v4

Authors: Nicolas Billerey , Imin Chen , Luis Dieulefait , Nuno Freitas

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 $p$ .

Keywords

generalized Fermat equations Frey elliptic curves modular method multi-Frey technique totally real fields level raising exponent bounds

Cite

@article{arxiv.1703.06530,
  title  = {A multi-Frey approach to Fermat equations of signature $(r,r,p)$},
  author = {Nicolas Billerey and Imin Chen and Luis Dieulefait and Nuno Freitas},
  journal= {arXiv preprint arXiv:1703.06530},
  year   = {2024}
}

Comments

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

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

Abstract

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