Contact information
Name:
C. Jayne Trent
Email:
trentj@lafayette.edu
When:
Thursday, March 28, 2024 - 4:15pm - 5:30pm
Where:
Pardee 217
Presenter:
Alex Barrios, University of St. Thomas
Price:
Free and there will be snacks!
Let a, b, and c be relatively prime positive integers such that a+b=c. How does c compare to rad(abc), where rad(n) denotes the product of the distinct prime factors of n? According to the explicit abc conjecture, it is always the case that c is less than the square of rad(abc). This simple statement is incredibly powerful, and as a consequence, one gets a (marginal) proof of Fermat's Last Theorem for an exponent n greater than 5.
In this talk, we introduce Masser and Oesterlé's abc conjecture and discuss some of its consequences, as well as some of the numerical evidence for the conjecture. Along the way, we will discuss recent work from the PRiME 2022 summer research group on abc triples of the form (1, c - 1, c). We will then introduce elliptic curves and see that the abc conjecture has an equivalent formulation in this setting, namely, the modified Szpiro conjecture. We conclude the talk by discussing a recent result that establishes the existence of sharp lower bounds for the modified Szpiro ratio of an elliptic curve that depends only on its torsion structure.
Sponsored by:
Department of Mathematics