W(2;3,21) = 416 - a van der Waerden number conjectured by Kullmann in 2010, now proved with a Lean 4 kernel-checked proof [R]
The van der Waerden number W(2;3,21) = 416, conjectured by Kullmann in 2010, has now been formally proved using a Lean 4 kernel-checked proof. This proof, generated by Hupyy's neuro-symbolic system LPM (Large Proof Model), involves 209 cases and 38,661,140 leaf formulas. Each refutation was verified by cake_pr, an LRAT checker verified in CakeML, ensuring that every red-blue colouring of {1, ..., 416} contains a red 3-term or a blue 21-term arithmetic progression.
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PublishedOffset at this time: UTC+0Oct 29, 2026, 20:46 UTC
IngestedOffset at this time: UTC+0Oct 29, 2026, 23:01 UTC
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- Oct 29, 2026, 20:46
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- Oct 29, 2026, 23:01
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w(2;3,21) = 416 - today proved with a Lean 4 kernel- checked proof (preprint + 13.6 GB evidence package).
Every red-blue colouring of {1, ..., 416} contains a red 3-term or a blue 21-term arithmetic progression, and some colouring of {1, ..., 415} contains neither.
Kullmann conjectured the value in 2010; DeepMind's Formal Conjectures list still carried it as open this year.
The proof is a Lean 4 theorem over 209 cases and 38,661,140 leaf formulas, each refutation checked by cake_pr (the LRAT checker verified in CakeML) and bound by digest to its Lean term.
It was produced with LPM (Large Proof Model), a neuro-symbolic system from Hupyy; nothing counted unless the Lean kernel or cake_pr accepted it.
Paper: https://doi.org/10.5281/zenodo.23776306
Evidence package (Lean sources, checker records, replay kit): https://doi.org/10.5281/zenodo.23749222
To check: download it and run 'shasum -a 256 -c SHA256SUMS'