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arXiv · 1709.01022

A conjecture of Erdos, supersingular primes and short character sums

Michael A. Bennett·Samir Siksek
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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.

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Michael A. Bennett, Samir Siksek. 2017-09-04. A conjecture of Erdos, supersingular primes and short character sums. https://arxiv.org/abs/1709.01022

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