Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 373

claims/: The 1 claim page of Problem 373, one per claimant's result; the problem's standing derives from them.

Statement. Show that the equation

n! = a_1!a_2!\cdots a_k!,

with $n-1>a_1\geq a_2\geq \cdots \geq a_k\geq 2$ , has only finitely many solutions.

Status. Open, the site's label (OPEN, which the site explains as open and not resolvable by a finite computation; page last edited 29 January 2026, accessed 2026-10-07). No claim settles the question. The one claim page, Luca's finiteness under the abc conjecture (claim page), is conditional and settles no standing; the Current assessment records the other conditional reduction and the unconditional results.

Source. erdosproblems.com/373, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #373, https://www.erdosproblems.com/373.

References.

[BhRa10] Bhat, K. Dzh. and Ramachandra, K., A remark on factorials that are products of factorials. Mat. Zametki 88 (2010), no. 3, 350-354.
[Ca94] C. Caldwell, The Diophantine equation $A!B!=C!$ . J. Recreat. Math. (1994), 128-133.
[Er76d] Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.
[Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. 16 (1993), 333--350, doi:10.1080/16073606.1993.9631741. The site's key for this problem, which locates nothing in this source: the survey restricts itself to graph theory (abstract, printed p. 333) and contains no passage on factorials or on this equation, so the site's key attaches to this paper a result it does not hold, and the paper carrying the bound the commentary credits under it is not identified here. Library home: erdos_1993_my_favorite_solved_unsolved_problems_graph_theory.
[Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0. Section B23 "Equal products of factorials", printed p. 123, where the book states the equation, the trivial family, Hickerson's nontrivial solutions, the searches to 18160 and $10^6$ , and Erdős's observation that $P(n(n+1))/\log n\to\infty$ would leave only finitely many nontrivial solutions. Library home: guy_2004_unsolved_problems_number_theory.
[Ha19] Habsieger, Laurent, Explicit bounds for the Diophantine equation $A!B!=C!$ . Fibonacci Quart. (2019), 21-28.
[Lu07b] Luca, Florian, On factorials which are products of factorials. Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 3, 533-542, doi:10.1017/S0305004107000308.

Formalization. Statement in formal-conjectures, pinned to the repository's revision of 2026-10-06, where erdos_373, the finiteness of the set of nontrivial solutions, is tagged open and carries no formal proof. The file also states, as variants without proof, the two implications from the hypotheses $P(n(n+1))/\log n\to\infty$ and $P(n(n-1))>4\log n$ for all large $n$ , tagged solved as literature results, and, tagged open, Hickerson's conjecture that $16!=14!\,5!\,2!$ is the largest solution and Surányi's conjecture for $k=2$ . The community database lists the problem as unformalized. A statement file is not a formalization.

Current assessment

The standing judges the site's formulation of 2026-09-04 above: the nontrivial solutions of $n!=a_1!\cdots a_k!$ are finitely many, where the condition $a_1<n-1$ excludes the trivial solutions in which $n=a_2!\cdots a_k!$ and $a_1=n-1$ . The question is open. No claim settles it; the one claim page is conditional.

Conditional results. Luca [Lu07b] proved finiteness under the abc conjecture, a refereed result recorded as an accepted conditional claim on its claim page, which settles no standing. Erdős [Er76d] proved (Theorem 2 of that paper) that if $P(n(n-1))>4\log n$ for all large $n$ , where $P(m)$ is the largest prime factor of $m$ , then for all large $n$ the equation has only trivial solutions, so finiteness follows; the site's commentary also records that $P(n(n+1))/\log n\to\infty$ would suffice, an observation Guy's section B23 [Gu04] attributes to Erdős, and the growth of $P(n(n+1))$ is Problem 368. Erdős's reduction gets no claim page: it is a conditional theorem whose hypothesis is itself open, it decides nothing unconditionally, and its source is a conference proceedings paper with no refereeing evidence while the site labels the problem OPEN, so a page would carry no acceptance evidence; the reduction is recorded here instead.

Unconditional results. Luca [Lu07b] proved that the set of $n$ with a nontrivial solution has asymptotic density zero; the site's commentary states the bound $\exp(f(x)\log x/\log\log x)$ on the number of such $n\le x$ for any $f(x)\to\infty$ . The site credits Erdős, under its key [Er93], with the bound $a_1\ge n-5\log\log n$ for $k=2$ and with the wish for $a_1\ge n-o(\log\log n)$ ; the site's reference record resolves that key to the 1993 graph-theory survey, which contains no such passage (see References), so the paper holding the bound is not identified here. Bhat and Ramachandra [BhRa10] replace the $5$ by $(1+o(1))/\log2$ and prove the bound for every $k\ge2$ (card). Hickerson's conjecture, reported in [Er76d], is that the only nontrivial solutions are $9!=2!\,3!\,3!\,7!$ , $10!=6!\,7!$ , $10!=3!\,5!\,7!$ and $16!=14!\,5!\,2!$ ; Surányi conjectured earlier that $6!\,7!=10!$ is the only nontrivial solution with $k=2$ . The computations of Caldwell [Ca94] and Habsieger [Ha19] (card) find no solution of $n!=a_1!\,a_2!$ other than $10!=6!\,7!$ for $n\le10^{3000}$ , as the site's commentary records; Habsieger's Theorem 1.4 states the stronger form, that every other solution has $a_1\ge10^{3000}$ . Guy's section B23 [Gu04] states the equation, Hickerson's solutions and the searches to 18160 and $10^6$ .

Dated search scope (2026-10-07): the site's problem page and commentary, its discussion thread and its proof-claims tab, which carries no claim; the community database's entry; the formal-conjectures statement file at the pinned revision; the publisher's record of Luca's paper; and the library cards for [Er76d], [Er93], [Gu04], [BhRa10] and [Ha19]. No other claim on the problem was found. Nothing on this page is independently reviewed.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (18)

Luca's finiteness under the abc conjecture problems/factorials_binomials/E0373/claims Factorials and Binomial Coefficients diophantine_problems/erdos_1976_problems_results_number_theoretic_properties_consecutive Theorem 2: P(n(n−1)) > 4 log n forces only trivial solutions of n! = a!b! and n! = a_1!...a_r!diophantine_problems/luca_2014_squares_factorials_products_factorials Theorem 3 (p. 6): under the explicit abc conjecture, a_2 is bounded in a_2!...a_t! = m(m+1)...(m+k-1)Theorem 4 (p. 6): unconditional bounds for a_2 in a_2!...a_t! = m(m+1)...(m+k-1)extremal_graph_theory/erdos_1993_my_favorite_solved_unsolved_problems_graph_theory Factorials and Binomial Coefficients factorials_binomials/bhat_2010_remark_factorials_that_are_products_factorials Theorem (pp. 350-351, unnumbered): if n! is (a_1!...a_k!) b! with 1 < a_1 <= ... <= a_k <= b < n, then n - b < ((1+epsilon)/log 2) log log n for large n factorials_binomials/habsieger_2019_explicit_bounds_diophantine_equation Theorem 1.1 (p. 22): the smaller factorial in A!B! = C! is at most log2(B+1) + 2 log2 log(B+1) + 2.1221 Theorem 1.2 (p. 22): C - B <= log log(B+1)/log 2 + 1.819 in A!B! = C!Theorem 1.3 (p. 22): B - A > C - log(C+1)/log 2 - 3 log log(C+1)/log 2 - 3.9411 in A!B! = C!Theorem 1.4 (p. 22): every nontrivial solution of A!B! = C! other than (6,7,10) has B >= 10^3000 number_theory/guy_2004_unsolved_problems_number_theory

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