A conjecture of Erdős, supersingular primes and short character sums

Abstract

If $k$ is a sufficiently large positive integer, we show that the Diophantine equation $$ n ( n+d) \cdots (n+ (k-1)d ) = y^{\ell } $$ has at most finitely many solutions in positive integers $n, d, y$ and $\ell $, with gcd$(n,d)=1$ and $\ell \geq 2$. Our proof relies upon Frey-Hellegouarch curves and results on supersingular primes for elliptic curves without complex multiplication, derived from upper bounds for short character sums and sieves, analytic and combinatorial.