Pedro-Jose Cazorla Garcia : The generalised Lebesgue-Ramanujan-Nagell equation

Posted by Alina Carmen Cojocaru , part of the Number Theory Seminar.

At

May 2, 2023, 8:45 a.m.

In

612 SEO

Abstract

The Lebesgue-Ramanujan-Nagell equation $x^2 + D = y^n$ has been studied extensively by number theorists during the last century, both for its intrinsic interest and as a generalisation to Catalan's conjecture. With the advent of the modular methodology developed by Wiles, Ribet and others and employed in the proof of Fermat's last theorem, along with the evolution of computational number theory techniques, it is now feasible to consider the generalised Lebesgue-Ramanujan-Nagell equation $C_1 x^2 + C_2 = y^n$. In this talk, we will discuss a variety of techniques which allow us to solve the aforementioned equation in the range $1 \leq C_1, C_2 \leq 20$, involving the modularity of Galois representations and Thue equations.

The seminar, including the question and discussion part, will take place between 8:45 and 9:45.