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

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

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

Number Theory 2025-10-16 v1

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

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.

Keywords

Fermat-type equation unit sieve Frey modular method level raising Galois representations

Cite

@article{arxiv.2510.13773,
  title  = {Revisiting the Fermat-type equation $x^{13} + y^{13} = 3z^7$},
  author = {Nicolas Billerey and Imin Chen and Lassina Dembélé and Luis Dieulefait and Nuno Freitas},
  journal= {arXiv preprint arXiv:2510.13773},
  year   = {2025}
}

Comments

9 pages

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