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

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 $p$.

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 for all integers , and all integers which are not a multiple of , 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 . We also give a number of results for the equations , where , 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 modulo .

Paper Structure

This paper contains 17 sections, 28 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Our Diophantine results
  3. Overview of the multi-Frey modular method
  4. The image of inertia argument
  5. A multi-Frey approach to the equation $x^{5} + y^{5} = 3 z^p$
  6. The modular method over $\mathbb{Q}$
  7. The modular method over $\mathbb{Q}(\sqrt{5})$
  8. Bounding the exponent
  9. Proof of Theorem \ref{['T:main5']}
  10. Partial results for $x^5 + y^5 = d z^p$ with $d = 1, 2$
  11. Proof of Theorem \ref{['T:main5a']}
  12. Proof of Theorem \ref{['T:main5b']}
  13. A result on the second case
  14. A multi-Frey approach to the equation $x^{13} + y^{13} = 3z^p$
  15. The modular method over $\mathbb{Q}(\sqrt{13})$
  16. ...and 2 more sections

Key Result

Theorem 1

For all primes $p$, there are no non-trivial primitive solutions to

Theorems & Definitions (51)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 3.3
  • Lemma 1
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • ...and 41 more