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Claim. R. Marszałek, On the product of consecutive elements of an arithmetic progression, Monatsh. Math. 100 (1985), no. 3, 215--222. Using the method of Erdős and Selfridge, the paper proves that the product $(n+d)(n+2d)\cdots(n+kd)$ is not a perfect power once $k$ is large in terms of $d$: as the zbMATH review (Zbl 0582.10011) states the bound, every length $k$ for which the product is a power satisfies

$$k\le\max\{3\cdot10^4,\ \tfrac32\exp[d(d+2)(d+1)^{1/3}]\}.$$

Bennett, Bruin, Győry and Hajdu (Proc. London Math. Soc. (3) 92 (2006), Section 1) and Bennett and Siksek (Ann. of Math. 191 (2020), Section 1) cite the paper for the coprime equation $n(n+d)\cdots(n+(k-1)d)=y^\ell$, $\gcd(n,d)=1$, in the case of fixed $d$. The library holds no copy of the paper. For each $d$ this settles all but finitely many lengths of Problem 672 in the negative.

Covers. For each fixed $d\ge1$, every length $k$ above the bound displayed, with every exponent $\ell\ge2$. Not covered: the lengths up to the bound, and no length is settled for all $d$ at once.

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

Acceptance. Refereed: Monatshefte für Mathematik 100 (1985), no. 3; the Crossref record dates the issue to September 1985, filled to the first of the month for this page's name. The site's commentary credits the result, but the site labels the problem VERIFIABLE, an open label, so the commentary is not reviewed evidence.