Tuesday, 28th Feb 2017
V205, Mathematics Building
(University of British Columbia)
Erdos-Selfridge and Supersingularity
In 1975, culminating more than 40 years of published work by Paul Erdos on the problem, he and John Selfridge proved that the product of consecutive integers cannot be a nonzero perfect power. Their proof was a remarkable combination of elementary and graph theoretic arguments. Subsequently, Erdos conjectured that this result can be generalized to a product of consecutive terms in an arithmetic progression, under certain basic assumptions. In this talk, we discuss joint work with Samir Siksek in the direction of proving Erdos' conjecture. Our approach is via techniques based upon the modularity of Galois representations, bounds for the number of supersingular primes for elliptic curves, and analytic estimates for Dirichlet character sums.