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Revisiting the Fermat-type equation $x^{13} + y^{13} = 3z^7$

Nicolas Billerey, Imin Chen, Lassina Dembélé, Luis Dieulefait, Nuno Freitas

TL;DR

The paper solves the Fermat-type equation $x^{13}+y^{13}=3z^7$ by completing the $p=7$ case of the modular method with a new modular unit sieve, the multi-Frey approach, and level-raising techniques, together with computations of systems of eigenvalues modulo $7$ over the totally real cubic field $K$ of $ obreak{ ext{Q}}( obreak{ ext{ extstyle aisebox{0.5ex}{$ ext{13}$}}})$. By forcing level-raising primes and proving reducibility of the corresponding mod $7$ representations, the authors eliminate all potential irreducible forms, culminating in a contradiction unless no non-trivial primitive solutions exist; in particular, they prove that any primitive solution must satisfy $13igm|a+b$ and $4igm|a+b$, which, combined with the modular eliminations, yields the nonexistence of non-trivial primitive solutions. The work fixes a prior bug in the unit sieve, introduces new techniques for handling small exponents, and advances the toolbox of the modular method for Fermat-type equations, with potential implications for related Diophantine problems and Chabauty-style approaches. The results underscore the power of combining unit constraints, level-raising phenomena, and detailed Galois-representation analysis over totally real fields in ruling out sporadic small-exponent cases.

Abstract

We solve the Fermat-type equation \[ x^{13} + y^{13} = 3 z^7, \qquad \gcd(x,y,z) = 1 \] combining a unit sieve, the multi-Frey modular method, level raising, computations of systems of eigenvalues modulo 7 over a totally real field, and results for reducibility of certain Galois representations.

Revisiting the Fermat-type equation $x^{13} + y^{13} = 3z^7$

TL;DR

The paper solves the Fermat-type equation by completing the case of the modular method with a new modular unit sieve, the multi-Frey approach, and level-raising techniques, together with computations of systems of eigenvalues modulo over the totally real cubic field of ext{13}. By forcing level-raising primes and proving reducibility of the corresponding mod representations, the authors eliminate all potential irreducible forms, culminating in a contradiction unless no non-trivial primitive solutions exist; in particular, they prove that any primitive solution must satisfy and , which, combined with the modular eliminations, yields the nonexistence of non-trivial primitive solutions. The work fixes a prior bug in the unit sieve, introduces new techniques for handling small exponents, and advances the toolbox of the modular method for Fermat-type equations, with potential implications for related Diophantine problems and Chabauty-style approaches. The results underscore the power of combining unit constraints, level-raising phenomena, and detailed Galois-representation analysis over totally real fields in ruling out sporadic small-exponent cases.

Abstract

We solve the Fermat-type equation combining a unit sieve, the multi-Frey modular method, level raising, computations of systems of eigenvalues modulo 7 over a totally real field, and results for reducibility of certain Galois representations.
Paper Structure (6 sections, 6 theorems, 15 equations, 1 table)

This paper contains 6 sections, 6 theorems, 15 equations, 1 table.

Table of Contents

  1. Introduction
  2. A modular unit sieve
  3. Forcing level raising primes
  4. Eliminating newforms modulo 7
  5. Newforms with reducible mod representations
  6. Proof of Theorem \ref{['T:reducibleSystems']}

Key Result

Theorem 1.1

There are no non-trivial primitive integer solutions to the equation

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Theorem 4.1
  • Remark 4.2
  • Proposition 5.1
  • proof
  • ...and 5 more