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.
时间与来源
时间显示为 UTC
显示时区：UTC
本地时区尚不可用，暂时显示 UTC。
发布当时偏移：UTC+02026年10月29日 20:46 UTC
收录当时偏移：UTC+02026年10月29日 23:01 UTC
- 发布
- 2026年10月29日 20:46
- 收录
- 2026年10月29日 23:01
- 来源类型
- 开发者社区
- 档位
- 社区
- 信源状态
- 正常
档位是按信源手工设定的编辑判断，不是逐条打分。
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'