Sendov’s 1958 conjecture on polynomial roots proved with GPT-5.6 Pro
Tao simplifies and formalises it
Result confirmed
Importance: major (4 of 5)The takeaway
Lech Mazur posted a computer-assisted proof, generated with GPT-5.6 Pro, of Sendov’s conjecture for all degrees: if every root of a polynomial lies in the closed unit disk, each root is within distance 1 of a critical point.
Status
- Claim
Result confirmed
- Our reporting
- High confidence
- Verification
- Lean-verified
- Importance
- Major (4 of 5)
- Last verified
- 19 October 2026
Your AI and this story
- GPT-6 Astra97 days after its cutoff
- Claude Opus 5.536 days after its cutoff
- Gemini 4 Hydrogen127 days after its cutoff
- Grok 4.766 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 36 days before it.
Key facts
- Conjecture from 1958; previously known for degree < 9 (Brown–Xiang) and for sufficiently large degree (Tao, 2020)
- Mazur’s preprint 5 Aug 2026 (some lists say 3 Aug); Tao’s digestion 12 Aug 2026
- Tao extended the method to the Phelps–Rodriguez conjecture
What happened
A non-academic used GPT-5.6 Pro to generate a computer-assisted proof covering the remaining degrees. Tao then digested it into a short argument based on the fundamental theorem of algebra and the Maclaurin inequality.
Why it matters
It is a classic, well-known conjecture closed by AI, with the leading expert on the problem verifying and formalising the result.
Sources
2 sources from 2 sites. Numbers match the chips in the text.
2 sources: 1 primary, 1 reaction
Primary
- Lech Mazur: Sendov conjecture proof (PDF)proofatlas.ai, paper
Reactions
- Terence Tao: A digestion of the proof of Sendov’s conjectureterrytao.wordpress.com, discussion
Changes
- Related: back-link to Lech Mazur’s Collatz result (same author and ProofAtlas harness)
- Filed