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arXiv · 2510.13773

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

Nicolas Billerey·Imin Chen·Lassina Dembélé·Luis Dieulefait·Nuno Freitas
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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.

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Nicolas Billerey, Imin Chen, Lassina Dembélé, Luis Dieulefait, Nuno Freitas. 2025-10-15. Revisiting the Fermat-type equation $x^{13} + y^{13} = 3z^7$. https://arxiv.org/abs/2510.13773

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