Third Portuguese Number Theory meeting

A perfect power is a number of the form \(x^a \) where \(x \geq 1\) and \(a \geq 2\) are integers. In 1931, S. S. Pillai conjectured that for any integer \(C\), the number of positive solutions \((x,y,a,b)\), with \(a \geq 2\) and \(b \geq 2\), 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.