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

A conjecture of Erdos, supersingular primes and short character sums