Second-order asymptotics for distinct factors of n! above n
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Let f(n) be the smallest possible largest factor in a factorization n!=a₁⋯aₖ with distinct integers n<a₁<⋯<aₖ. What is the second-order growth of f(n) beyond 2n?
Why it matters and what counts as progress
Why it matters: The second-order term distinguishes factorizations with the same leading 2n scale and asks for a sharp construction.
What counts as progress: Match upper and lower bounds for (f(n)−2n)log n/n or determine its limiting constant.
How the problem developed
- 1982 · Partial result [1]
Paul Erdős, Richard K. Guy and John L. Selfridge Established the order of growth f(n)−2n≈n/log n for the least possible largest distinct factor in a factorization of n! into factors exceeding n.
- 3 May 2026 · Lower bound [2]
Mausberg Reported the thirteen-layer lower-bound constant later appearing in the claimed limit, without a matching upper bound or asymptotic formula.
- 28 Jul 2026 · Proof claim [3]
Shouqiao Wang A manuscript claims that (f(n)−2n)log(n)/n tends to 4029639598/25970038185, adding a matching upper-bound construction.
AI contribution
A Proposed Solution to Erdős Problem 390
For the least possible largest distinct factor f(n) in n! = a₁⋯aₖ with n<a₁<⋯<aₖ, Wang’s manuscript claims that (f(n)−2n)log(n)/n tends to 4029639598/25970038185. Its new step is a matching upper-bound construction.
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