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Claim. Assume the abc conjecture. Then the equation $n!=a_1!a_2!\cdots a_k!$ with $n-1>a_1\ge a_2\ge\cdots\ge a_k\ge2$ has only finitely many solutions, the statement of Problem 373. This is the conditional theorem of F. Luca, On factorials which are products of factorials, Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 3, 533--542, whose abstract states that under the abc conjecture the equation has only finitely many nontrivial solutions and, unconditionally, that the set of $n$ for which it has a nontrivial solution has asymptotic density zero. The site's commentary (page last edited 29 January 2026) credits the paper with the conditional finiteness and states the unconditional result as a bound: the number of $n\le x$ that admit a nontrivial solution is at most $\exp(f(x)\log x/\log\log x)$ for any function $f(x)$ tending to infinity. Bhat and Ramachandra's note (bhat_2010_remark_factorials_that_are_products_factorials) cites the conditional result in the form that for all large $n$ every solution has $a_1=n-1$, that is, is trivial. The library has no card for Luca's paper; the citation is the publisher's record.

Hypothesis. The abc conjecture, as the abstract names it: in its standard form, for every $\epsilon>0$ there is a constant $K_\epsilon$ such that coprime positive integers $a+b=c$ satisfy $c<K_\epsilon\operatorname{rad}(abc)^{1+\epsilon}$. It is unproved, so the claim gives no unconditional answer.

Scope. The claim is conditional and settles no standing of the problem by itself. Unconditionally the problem is open; the problem page records the unconditional results, Luca's density-zero theorem among them, and the other conditional reduction, Erdős's, which has no claim page.

Depends on. Nothing in this wiki; the hypothesis is stated above.

Acceptance. Refereed: Mathematical Proceedings of the Cambridge Philosophical Society, volume 143, issue 3, pp. 533--542, issued November 2007 and published online 2007-11-01 by the publisher's record, which dates this page. The site labels the problem OPEN, so no curator acceptance is listed; the commentary's credit is context. Nothing here was checked by this project.