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.

中文翻译：

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Erdős、超奇异素数和短字符和的猜想

如果 $k$ 是一个足够大的正整数，我们证明丢番图方程 $$n (n+d) \cdots (n+ (k-1)d) = y^{\ell}$$ 至多具有有限多个正整数 $n, d, y$ 和 $\ell$ 的解，其中 $\operatorname{gcd}(n,d)=1$ 和 $\ell \geq 2$。我们的证明依赖于 Frey-Hellegouarch 曲线和椭圆曲线的超奇异素数结果，没有复杂的乘法，从短字符和和筛分的上限、解析和组合推导出来。