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Claim. K. Győry, L. Hajdu and Á. Pintér, Perfect powers from products of consecutive terms in arithmetic progression, Compos. Math. 145 (2009), no. 4, 845--864. Theorem 1.1 (printed p. 847) reads: "If $3<k<35$, the product of $k$ consecutive terms in a coprime positive arithmetic progression is never a perfect power." Here the progression is $x,x+d,\ldots,x+(k-1)d$ with positive $x,d$ and $\gcd(x,d)=1$. For $k\le11$ the paper takes the theorem from the earlier results; the new lengths $12\le k\le34$ come from its Theorems 1.2 and 1.3, proved by a reduction to ternary equations and a computer sieve. Library home: Theorem 1.1. These are the lengths $4\le k\le34$ of Problem 672.

Covers. Lengths $4\le k\le34$, every $d$ and every exponent $\ell\ge2$. Not covered: every length $k\ge35$.

Depends on. Bennett–Bruin–Győry–Hajdu and Győry–Hajdu–Saradha for the lengths $k\le11$.

Acceptance. Refereed: Compositio Mathematica 145 (2009), no. 4. The site's commentary credits the theorem, but the site labels the problem VERIFIABLE, an open label, so the commentary is not reviewed evidence.