Third Portuguese Number Theory meeting

A perfect power is a number of the form $x^a$ where $x\ge 1$ and $a\ge 2$ are integers. In 1931, S. S. Pillai conjectured that for any integer $C$, the number of positive solutions $(x,y,a,b)$, with $a\ge 2$ and $b\ge 2$, of the diophantine equation $y^a-x^b=C$ is finite. This conjecture amount to saying that the distance between two consecutive perfect powers tends to infinity. In 2022, we obtained all solutions of $2^a-p^b=2^c-1$, when $p=F_n$ is a Fermat prime. In this talk, we study the cases when $y=2$, $C=2^c-1$, for an integer $c$ and an odd number $x$. As an application we also study generalizations of the Euclides-Euler Theorem for certain even $\alpha$-perfect numbers. We will use the general method developed by Styer to solve exponential diophantine equations which formalizes and extends a method used by Guy, Lacampagne, and Selfridge.

I will explain a new approach to generalized Fermat equations of signature (r,r,p) based on multi-Frey techniques and ideas from Darmon’s program.

Beginning in the 80s with the celebrated work of Mazur, Tate and Teitelbaum, the study of exceptional zeros for p-adic L-functions has become a very fruitful area in number theory. In this talk, we begin by giving a historical survey of several applications of this theory, which include certain cases of the p-adic Birch and Swinnerton-Dyer conjecture and the Gross--Stark conjectures. We connect this with a result obtained during my PhD in a joint work with V. Rotger, and which can be seen as a Gross--Stark formula for the adjoint of a weight one modular form. Finally, we describe a tantalizing connection between our work and a deep conjecture of Harris and Venkatesh, which explains the presence of the same system of Hecke eigenvalues in multiple degrees of cohomology.