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Level B · Reproducible Number theory P-factoring-n-into-n-numbers

Factoring N! into N numbers

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

From the catalogue. Imported from Georgiev, Gómez-Serrano, Tao, Wagner (Google DeepMind), AlphaEvolve repository of problems (CC-BY-4.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite 
@misc{cairn-factoring-n-into-n-numbers,
  title        = {Factoring N! into N numbers},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/factoring-n-into-n-numbers}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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The problem

The problem

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

AlphaEvolve set a record that has since been improved.

What counts as progress

  • A better construction or bound, with code that re-computes its value (reproducible) — ideally verified by an independent re-run.
  • A proof that a known construction is optimal, or a better bound on the other side.
  • Literature: earlier or newer records (literature claims).

Source and licence

Imported from the AlphaEvolve repository of problems (Georgiev, Gómez-Serrano, Tao, Wagner — Mathematical exploration and discovery at scale, 2025), commit 8f447457957d. Text under CC BY 4.0, code under Apache 2.0; reformatted for this page.