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Claim. K. Győry, L. Hajdu and N. Saradha, On the Diophantine equation $n(n+d)\cdots(n+(k-1)d)=by^l$, Canad. Math. Bull. 47 (2004), no. 3, 373--388. For positive integers $n,d,y,b$, integers $k,\ell\ge2$, $\gcd(n,d)=1$, $P(b)\le k$ and $b$ free of $\ell$th powers (printed p. 373), Theorem 1 (printed p. 374) states that the equation has no solution when $k=4$ or $5$ and $b=1$: a product of four or five consecutive positive terms of a coprime arithmetic progression is never a perfect power. The case $\ell=2$ is taken from Euler ($k=4$) and from Obláth ($k=5$). Bennett, Bruin, Győry and Hajdu (2006, printed pp. 273--274) found the paper's argument for $\ell=3$ invalid and corrected it in their Section 5, so the theorem holds with that corrected proof. Library home: gyory_2004_diophantine_equation. These are the lengths $k=4$ and $k=5$ of Problem 672.

Covers. Lengths $k=4$ and $k=5$, every $d$ and every exponent $\ell\ge2$. Not covered: every length $k\ge6$.

Depends on. Bennett–Bruin–Győry–Hajdu (the corrected proof for $\ell=3$) and Obláth (the case $k=5$, $\ell=2$).

Acceptance. Refereed: Canadian Mathematical Bulletin 47 (2004), no. 3. The site's commentary credits the theorem, but the site labels the problem VERIFIABLE, an open label, so the commentary is not reviewed evidence.