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Claim. R. Obláth, Über das Produkt fünf aufeinander folgender Zahlen in einer arithmetischen Reihe, Publ. Math. Debrecen 1 (1950), 222--226. The paper proves that the product $n(n+d)(n+2d)(n+3d)(n+4d)$ of five consecutive terms of an arithmetic progression of positive integers with $\gcd(n,d)=1$ is never a perfect square, extending Euler's theorem for four terms. The statement is given as Győry, Hajdu and Saradha record it (Canad. Math. Bull. 47 (2004), printed p. 373, their reference [11], which "extended this result to the case $k=5$"; their proof of Theorem 1 takes the case $\ell=2$, $k=5$ from it) and as Bennett, Bruin, Győry and Hajdu record it (Proc. London Math. Soc. (3) 92 (2006), Section 1, their references [26] and [27]). The library holds no copy of the paper. This is the case $(k,\ell)=(5,2)$ of Problem 672. The site credits the case to Obláth under its key [Ob51], which names his 1951 note in J. Indian Math. Soc.; the 2004 paper places the case $k=5$ in this 1950 paper. The other cases the site credits to Obláth have $k=3$ and lie outside the question.

Covers. Length $k=5$ with exponent $\ell=2$ (and so every even exponent). Not covered: $k=5$ with odd exponents, and every other length.

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

Acceptance. Refereed: Publicationes Mathematicae Debrecen 1 (1950). The site's commentary credits the case, but the site labels the problem VERIFIABLE, an open label, so the commentary is not reviewed evidence. The case is also part of the later Győry–Hajdu–Saradha theorem, whose proof for $\ell=2$ cites this paper.