Number Theory Seminar
Pedro-Jose Cazorla Garcia
University of Manchester
The generalised Lebesgue-Ramanujan-Nagell equation
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.
Tuesday May 2, 2023 at 8:45 AM in 612 SEO