Sylvester’s 1879 conjecture on sums of two cubes is fully proved
Burungale and Tian settle the last case (p ≡ 8 mod 9) after Hongbo Yin’s proof of the 4, 7 cases; both later papers disclose AI use
Awaiting review
Importance: major (4 of 5)The takeaway
On Sept 14, 2026 Ashay Burungale (UT Austin) and Ye Tian (AMSS, Chinese Academy of Sciences) posted “A proof of Sylvester’s conjecture” (arXiv 2609.14893).
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 10 October 2026
Your AI and this story
- GPT-6 Astra137 days after its cutoff
- Claude Opus 5.576 days after its cutoff
- Gemini 3.8 Flash167 days after its cutoff
- Grok 4.7106 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 76 days before it.
Key facts
- Conjecture (Sylvester, 1879): every prime p ≡ 4, 7, 8 (mod 9) is a sum of two rational cubes. Primes p ≡ 2, 5 (mod 9) are not (Pépin, Lucas, Sylvester). BSD predicts the conjecture because the curve E_p: y² = x³ + p²/4 has root number −1 exactly for these classes
- History: Elkies announced a proof for p ≡ 4, 7 in 1994 but never published details; Dasgupta–Voight proved the 4, 7 cases under an extra cubic-residue condition (2009, 2018); Yin proved a weaker 8-case result (Trans. AMS 2022)
- May 25, 2026: Hongbo Yin (Shandong University), ‘A proof of the 4,7 cases of Sylvester’s conjecture on cube sums’, arXiv 2605.25917. It uses recent progress on full BSD for rank-0 curves and the solution of the Unbounded Denominators Conjecture
- Sept 14, 2026: Ashay Burungale and Ye Tian, ‘A proof of Sylvester’s conjecture’, arXiv 2609.14893 (v2 Sept 15). For p ≡ 8 (mod 9) they prove E_p has analytic rank one, adapting an auxiliary Rankin–Selberg construction from their work on the rank-one converse for CM elliptic curves
- Burungale–Tian AI disclosure: ‘Use of artificial intelligence. ChatGPT and Claude tools were used for exploratory computations, editing and reorganization.’
- Oct 6, 2026: Hongbo Yin, ‘A concise proof of the 8 case of Sylvester’s conjecture’, arXiv 2610.08015 (10 pages). It avoids the Gross–Zagier formula and calls Burungale–Tian’s proof ‘complicated and hard to follow’
- Yin’s AI disclosure: ‘The ideas are due to the author while the details realization is due to AI. But the output of AI is ugly and not readily readable by human, so the final simplification, reorganization and writing are due to the author. The example in Section 6 is computed by AI.’ The AI tool is not named
What happened
Sylvester’s conjecture goes back to his 1879 work on ternary cubic equations. It says that every prime p ≡ 4, 7 or 8 (mod 9) can be written as a sum of two rational cubes. In modern terms, the elliptic curve E_p: y² = x³ + p²/4 should have positive rank, which the Birch and Swinnerton-Dyer conjecture predicts from its root number.
Hongbo Yin (Shandong University) posted a proof of the 4 and 7 cases on arXiv on May 25, 2026 (revised in June), before the June 2026 cutoff. On Sept 14, 2026 Ashay Burungale and Ye Tian posted “A proof of Sylvester’s conjecture”, which handles the last class, p ≡ 8 (mod 9), by proving that E_p has analytic rank one. Together the two papers prove the full conjecture. Burungale and Tian state that ChatGPT and Claude were used for exploratory computations, editing and reorganization.
On Oct 6 Yin posted a 10-page “concise proof” of the 8 case built on his May paper. He writes that the ideas are his, the detailed working-out was done by an unnamed AI whose output was “ugly and not readily readable by human”, and he rewrote it.
We found this through the arXiv sweep of Oct 10 (Yin’s Oct 6 paper). We found no press coverage or expert reactions. Affiliations from the papers: Burungale, University of Texas at Austin; Tian, Morningside Center / AMSS, Chinese Academy of Sciences.
Why it matters
This is one of the oldest named problems in Diophantine number theory, and the main proofs are human-led. AI shows up as a tool for exploratory computation and, in the shorter follow-up proof, for writing out the details. It shows that by autumn 2026 even researchers proving major results routinely disclose AI help.
What is disputed or not yet verified
| Verification | Preprints on arXiv; not yet peer-reviewed; no formal verification |
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Sources
4 sources from 2 sites. Numbers match the chips in the text.
4 sources: 4 primary
Primary
- arXiv 2609.14893: Burungale & Tian, A proof of Sylvester’s conjecturearxiv.org, paper
- arXiv 2605.25917: H. Yin, A proof of the 4,7 cases of Sylvester’s conjecture on cube sumsarxiv.org, paper
- arXiv 2610.08015: H. Yin, A concise proof of the 8 case of Sylvester’s conjecturearxiv.org, paper
- Dasgupta & Voight, Sylvester’s problem and mock Heegner points (background)ar5iv.labs.arxiv.org, paper
Changes
- Filed from the Oct 10 arXiv sweep (Yin 2610.08015 led to Burungale–Tian 2609.14893)