Factoring N! into N numbers

AlphaEvolve improved the best known bound; the record has since been surpassed. The problem remains open.

Problem
Problem 38 in the AlphaEvolve repository: Factoring N! into N numbers
Credit
AlphaEvolve (Google DeepMind) · LLM-guided evolutionary code search, with Georgiev, Gómez-Serrano, Tao and Wagner
Checked
Construction can be checked directlyPreprint (arXiv, 2025)
Timeline
Paper submitted 2025-11-03
Sources
Problem page (AlphaEvolve repository)Paper (arXiv:2511.02864)
PARTIALImproved bound · new construction

The problem

For a natural number NN, let C(N)C(N) be the largest quantity such that N!N! can be factored into NN factors that are greater than or equal to C(N)C(N) (see OEIS A034258). Establish upper and lower bounds on C(N)C(N) that are as strong as possible.

Statement quoted from problem 38 of the AlphaEvolve repository of problems (Google DeepMind, CC BY 4.0).

What AI did

AlphaEvolve improved the best known bound; the record has since been surpassed. The problem remains open. Status in the repository: Former new result, since improved upon. See Section 6.21 in "Mathematical Exploration and Discovery at Scale" paper. Records may have moved since; the repository's notes below list the latest bounds it knows of.

How it was checked

The results are explicit constructions or programs; the repository's notebooks include the verification code used to score them.

Limitations

An improved bound is progress, not a solution: the exact answer to the problem is still unknown. The authors report that results were better when experts set up and guided the search.

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