Problem 373

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.

Source: erdosproblems.com/373

Number theory
Factorials

Wiki page

Status

Open

No claim settles this problem.

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 (Luca, 2007)), is conditional and settles no standing; the Current assessment records the other conditional reduction and the unconditional results.

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: Erdős 1993.
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.
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.

Problem 373

Status

Proof claims

Discussion