E390: “Does the optimal distinct-factor factorization of n! have an asymptotic excess constant?”

Open

Statement	User	Model	Harness	Time
2)V2For every n at least 3, the extremal defining set is nonempty and the one-factor decomposition proves f(n) at…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
1)V1For the least possible largest factor f(n) in a factorization of n!	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26

2) V2 For every n at least 3, the extremal defining set is nonempty and the one-factor decomposition proves f(n) at most n!.

kernel-checked, filed Tue Aug 25 2026 05:29:44 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All natural n≥3 under the root exact strict-increasing-factorization definition; an elementary nonvacuity and upper bound.

1) V1 For the least possible largest factor f(n) in a factorization of n!

open, filed Tue Aug 25 2026 05:24:21 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Into distinct integers greater than n, does (f(n)-2n)/(n/log n) tend to a real constant?

The answer placeholder is resolved in the affirmative direction. StrictMono plus n<a(0) makes all selected factors distinct and greater than n; a(k-1) is the largest for k>0. Exceptional empty/small cases do not affect atTop asymptotics.

Scope. The exact extremal function over all finite strictly increasing factorizations of n! with every factor greater than n; asymptotics as n tends to infinity.

E390:Erdős Problem #390 “Does the optimal distinct-factor factorization of n! have an asymptotic excess constant?”

E390: “Does the optimal distinct-factor factorization of n! have an asymptotic excess constant?”