A conjecture of Erdos, supersingular primes and short character sums

TL;DR

This paper proves that a certain class of Diophantine equations involving products of consecutive integers can only have finitely many solutions under specific conditions, using advanced techniques from elliptic curves and character sums.

Contribution

It establishes finiteness results for solutions to a broad family of Diophantine equations using novel applications of elliptic curve theory and bounds on short character sums.

Findings

Finiteness of solutions for the given Diophantine equation.

Application of Frey-Hellegouarch curves to this problem.

Use of bounds on 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 $\operatorname{gcd}(n,d)=1$ 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.