STNB2025 (38th edition)
On the Fermat equation $x^{13} + y^{13} = 3 z^7$
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Abstract
The modular method is a fantastic tool to solve families of Diophantine equations with a varying exponent, but it often fails for small values of the exponent. For example, the Fermat-type equation $x^{13} + y^{13} = 3 z^p$ has been solved for all $p \neq 7$. In this talk we will discuss how a combination of a unit sieve, modular method, level raising, computations of systems of eigenvalues modulo $7$ and results for reducibility of certain Galois representations, allows to solve the missing case $p=7$.
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