Algebra and Number Theory Seminar: Integral solutions of Generalized Fermat equations

Abstract: Since the resolution of Fermat's Last Theorem by Wiles/Taylor in 1993 the so-called modular method has been applied to solve both other classes of, but also specific instances of, Diophantine equations with a similar form, many of these Diophantine problems are tricky and simply not accessible by other approaches. Notably this includes trying to show that in all but finitely many cases we have only the trivial integral solutions for the generalized Fermat equations $x^{a}+ y^{b}= z^c$  for $a,b,c \in \mathbf N$ when $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}< 1$, this is sometimes known as the Fermat-Catalan conjecture, the version with coefficients is also of interest.

I'll report on recent and ongoing work with Dahmen and Freitas resolving some of the smallest unknown cases of this conjecture, those of the equations $x^{13}+ y^{13}= z^p$ for $p = 5, 7$.
This involves a combination of several Diophantine techniques, including the modular method, descent, variants of Chabauty's method and Chabauty over number fields, the unit sieve, and Mordell-Weil sieves.
I'll explain these techniques and how they can be fruitfully applied for this problem, but also more generally for other Diophantine equations.