The Fermat equation x^13+y^13=3z^7

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 $x^13+y^13=3z^p$ has been
solved for all $p\ne 7$. In this talk, we will discuss how a combination of a
unit sieve, the modular method, level raising and computations of systems
of eigenvalues modulo $7$, and results for reducibility of certain Galois
representations, allows us to solve the missing case $p=7$.