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Room P3.10, Mathematics Building

Nuno Freitas, Universitat de Barcelona

From Fermat's Last Theorem to some generalized Fermat equations

The proof of Fermat's Last Theorem was initiated by Frey,
Hellegouarch, Serre, further developed by Ribet and ended with
Wiles' proof of the Shimura-Tanyama conjecture for semi-stable
elliptic curves. Their strategy, now called the modular approach,
makes a remarkable use of elliptic curves, Galois representations
and modular forms to show that ${a}^{p}+{b}^{p}={c}^{p}$ has no solutions,
such that $(a,b,c)=1$ if $p\ge 3$. Over the last 17 years, the
modular approach has been continually extended and allowed people
to solve many other Diophantine equations that previously seemed
intractable. In this talk we will use the equation ${x}^{p}+{2}^{\alpha}{y}^{p}={z}^{p}$ as the motivation to introduce informally the
original strategy ($\alpha =0$) and illustrate one of its first
refinements (for $\alpha =1$). Then we will discuss some further
generalizations that recently led to the solution of equations of
the form ${x}^{5}+{y}^{5}=d{z}^{p}$.