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Pollack and Shapiro are credited with the case $y=1$, $k=4$ of Problem 399: $n!+1$ is never a fourth power, that is, $n!=x^4-1$ has no solution. Their paper, The next to last case of a factorial Diophantine equation, cites the paper of Erdős and Obláth, whose Satz 3 excludes coprime differences of fourth powers only for sufficiently large $n$.

Covers. $n!=x^4-1$ has no solution. This is the site's account. Erdős and Graham's monograph, the problem's source (p. 77), reports more: that the paper shows there are no solutions with $\gcd(x,y)=1$ and $k=4$, which would close the coprime case for every $k>2$. The site's own page gave that wider account in its archived copy of 7 November 2024 and the narrower one from its copy of 7 April 2025, both linked above. Neither the paper nor a review of it stating its theorem is held, so the wider report is recorded as unconfirmed.

Depends on. No page of this wiki.

Acceptance. Refereed: R. M. Pollack and H. N. Shapiro, The next to last case of a factorial diophantine equation, Comm. Pure Appl. Math. 26 (1973), no. 3, 313–325. The site's curator credits the result in the commentary, but the site's label settles the problem through Barfield's counterexample and not through this result, so reviewed is not listed. The formal-conjectures file states the site's version as erdos_399.variants.pollack_shapiro with sorry, which is not a formalization.