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Claim. Theorem 1 of Erdős and Selfridge 1975: the equation $(n+1)(n+2)\cdots(n+k)=x^l$ has no solution in integers with $k\geq 2$, $l\geq 2$ and $n\geq 0$, so a product of two or more consecutive positive integers is never a perfect power. The proof passes through their Theorem 2, which finds a prime $p\geq k$ dividing the product to a power not divisible by $l$.

Covers. The case $r=1$ of Problem 930: for a single interval of positive integers, $k=2$ already works.

Standing. The claim is accepted on refereed evidence: the theorem was published in Illinois J. Math. 19 (1975), no. 2, 292-301. The site's remark crediting the case $r=1$ to this paper is not counted as review, since the site labels the problem OPEN.

Depends on. No page of this wiki.