Pure Math Grad Student Colloquium

Nicolas Banks, Department of Pure Mathematics, University of Waterloo

"ABC Conjecture and Fermat's Last Theorem for Polynomials"

The ABC conjecture, posed in the 1980s, asserts (roughly) that if integers A and B are divisible by large powers of small primes, then their sum C=A+B is usually divisible by small powers of large primes. It has a number of important consequences, both known and unknown, such as Roth's theorem on approximations of rational numbers, Faltings' theorem on rational points on projective curves, and the Fermat-Catalan conjecture, which generalizes Fermat's Last Theorem. Despite a claimed proof by Mochizuki in the last decade, the conjecture is still considered unproved by the general mathematical community.

There is an analogous result for polynomials called the Mason-Stothers theorem. Oddly, this theorem is far easier to prove. In fact, an elegant elementary proof was discovered by a high school student in 2000. We present this proof, then use it to prove an analogue of Fermat's Last Theorem for polynomials.

Zoom meeting: https://uwaterloo.zoom.us/j/95669082908?pwd=L0NhbVZBakFGQXUycUZka01CL2lsQT09