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Claim. M. A. Bennett, N. Bruin, K. Győry and L. Hajdu, Powers from products of consecutive terms in arithmetic progression, Proc. London Math. Soc. (3) 92 (2006), no. 2, 273--306. Theorem 1.1 (printed p. 274) reads: "The product of $k$ consecutive terms in a coprime positive arithmetic progression with $4 \leq k \leq 11$ can never be a perfect power." A coprime progression is $n,n+d,\ldots,n+(k-1)d$ with $\gcd(n,d)=1$. The theorem follows from the paper's Theorem 1.2 on $n(n+d)\cdots(n+(k-1)d)=by^\ell$ for prime $\ell$, and its Section 5 corrects the argument of Győry, Hajdu and Saradha for $\ell=3$. Library home: bennett_2006_powers_products_consecutive_terms_arithmetic_progression. These are the lengths $4\le k\le11$ of Problem 672. The site's second credit to the paper, for $k$ large in terms of the number of prime factors of $d$, is Corollary 1.6 (printed p. 276), which gives finitely many solutions and settles no instance.

Covers. Lengths $4\le k\le11$, every $d$ and every exponent $\ell\ge2$. Not covered: every length $k\ge12$.

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

Acceptance. Refereed: Proceedings of the London Mathematical Society (3) 92 (2006), no. 2 (received 27 September 2004, revised 7 June 2005). The site's commentary credits the theorem, but the site labels the problem VERIFIABLE, an open label, so the commentary is not reviewed evidence.