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Claim. P. Erdős and J. L. Selfridge, The product of consecutive integers is never a power, Illinois J. Math. 19 (1975), no. 2, 292--301. Theorem 1 (printed p. 292) reads: "The product of two or more consecutive positive integers is never a power." Equivalently, $(n+1)(n+2)\cdots(n+k)=x^\ell$ has no solution with $k\ge2$, $\ell\ge2$ and $n\ge0$. A progression with $d=1$ is coprime for every initial term, so for $d=1$ the product in Problem 672 is never a perfect power, for every length $k\ge4$. Library home: erdos_1975_product_consecutive_integers_is_never_power. The paper's Section 4 (printed p. 300) adds, without proof, that for each fixed $d$ there must be a threshold $t_d$ beyond which the product of $t$ terms is never a power; that remark is not part of this claim.

Covers. Common difference $d=1$, every length $k\ge4$ and every exponent $\ell\ge2$. Not covered: every $d\ge2$.

Depends on. Nothing in this wiki; the result rests on the cited paper.

Acceptance. Refereed: Illinois Journal of Mathematics 19 (1975), no. 2. The site's commentary credits the theorem, but the site labels the problem VERIFIABLE, an open label, so the commentary is not reviewed evidence.