MPIM
The Fermat equation $x^13+y^13=3z^7$MPIM
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Europe/Berlin
MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)
MPIM, Vivatsgasse, 7 - Lecture Hall
Max Planck Institute for Mathematics
120
Description
Number theory lunch seminar
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 $p=7$.