Artin–Davenport conjecture proved: every integral cubic form in 10+ variables has a nontrivial zero (Bonolis, Browning, Glas, Wang; key lemma via ChatGPT 6.0, Codex Lean auto-formalisation)
On Oct 6, 2026 Dante Bonolis, Tim Browning, Jakob Glas and Victor Y. Wang posted arXiv 2610.08226, a 64-page proof that every integral cubic form in ten or more variables has a nontrivial integer zero. The bound of 10 is sharp, and the best previous results were Davenport's 16 variables (1963) and Heath-Brown's 14 (2007). The authors say the structure and most new ideas are theirs (June 2024 – September 2026), but a key dimension bound (Lemma 3.8) came "after prompting ChatGPT 6.0". A Codex Lean auto-formalisation of the main theorem, modulo five literature inputs, is public. It is an unrefereed preprint.
Key facts
- Paper: 'The Artin–Davenport conjecture on cubic forms', Dante Bonolis, Tim Browning, Jakob Glas, Victor Y. Wang, arXiv 2610.08226 (math.NT), 6 Oct 2026, 64 pages
- Theorem 1.1: 'The Artin–Davenport conjecture is true': any integral cubic form in n ≥ 10 variables has a nontrivial integer zero
- History (paper): p-adic solubility for n ≥ 10 known since Demyanov (1950) and Lewis (1952); Davenport's circle method gave n ≥ 32 (1959), then n ≥ 16 (1963); Heath-Brown n ≥ 14 (2007). Heath-Brown (1983) and Hooley had settled smooth and isolated-singularity cases. Ten is optimal: a nonary norm-form example has only the trivial p-adic zero
- Method: geometric methods when the singular locus is large, otherwise a Kloosterman circle method with stratification of exponential sums (singular support: Beilinson, Hu–Yang, Raskin–Smith), a stratified geometric sieve and a cube-root Hensel lifting analysis
- AI statement (quote): 'After prompting ChatGPT 6.0, we were led to the improved bound dim I ⩽ ⌊(4n − 3)/3⌋ in Lemma 3.8 … ChatGPT 6.0 suggested an approach via the Galois descent procedure in Lemma 3.6'. Heath-Brown's private improvement to ⌊(3n − 3)/2⌋ would, the authors believe, only have handled eleven or more variables
- Other AI use (quote): 'ChatGPT 5.5, 5.6, 6.0 and 6.1 have been helpful with this process, as well as in proofreading our work, and conducting the Lean auto-formalisation'; the authors thank OpenAI's Mark Sellke for access to the Pro models in May 2026
- Lean: github.com/wangyangvictor/cubic-lean-mid, 'a Codex Lean auto-formalisation of Theorem 1.1 and its proof, modulo five inputs from the literature in elementary form' (not a complete formal proof)
- By-product: Theorem 1.3, a cubic hypersurface over Q whose singular locus has codimension at most ⌊(n−1)/3⌋ has a rational point
- Not in OpenAI's 6 Oct github.com/openai/math release (no cubic-form family in its catalogue)
Science result
- Field
- mathematics / number theory / Diophantine geometry
- Problem
- Artin–Davenport conjecture: integral cubic forms in at least 10 variables have nontrivial integer zeros
- Result
- Proved in full: n ≥ 10 suffices (previous record n ≥ 14, Heath-Brown 2007); 10 is sharp.
- AI system
- ChatGPT 6.0 (GPT-6), ChatGPT 5.5/5.6/6.1, OpenAI Codex
- Human role
- Human-led over two years; ChatGPT 6.0 supplied the improved incidence-variety dimension bound (Lemma 3.8) and the Galois-descent approach of Lemma 3.6; GPT models helped combine estimates and proofread; Codex produced a partial Lean auto-formalisation
- Verification
- Unrefereed preprint; partial Lean auto-formalisation (main theorem modulo five literature inputs)
- Status
- pending
- Why surprising
- A classical Diophantine problem that had moved from 16 to 14 variables in 44 years went straight to the sharp bound, with a frontier model supplying a lemma the authors say made ten variables reachable.
What happened
On 6 October 2026, Dante Bonolis (TU Graz), Tim Browning (ISTA and Imperial College London), Jakob Glas (Leibniz Universität Hannover) and Victor Y. Wang (Université Paris-Saclay and Academia Sinica) posted a proof of the Artin–Davenport conjecture: every cubic form with integer coefficients in ten or more variables has a nontrivial integer zero. Since the early 1950s it has been known that ten variables are enough for solutions over every p-adic field and that ten is the least such number. Over the integers, Davenport's circle method needed 32 variables in 1959 and 16 in 1963, and Heath-Brown brought it down to 14 in 2007. The new 64-page paper closes the gap.
The proof splits into cases. If the cubic hypersurface has a large singular locus, geometric arguments produce a rational point. Otherwise a Kloosterman form of the circle method is used, and the key new input is to control where square-root cancellation fails in the relevant exponential sums by a stratification based on singular support. The authors add a stratified geometric sieve, estimates for "cubefull" moduli and a cube-root Hensel lifting analysis.
The AI statement is specific. The authors write that the structure of the proof and most of the new ideas are theirs, developed from June 2024 to September 2026. A key step needed an upper bound on the dimension of an incidence variety attached to the Hessian. They had proved ⌊(3n−2)/2⌋, and Heath-Brown privately improved it to ⌊(3n−3)/2⌋, which they think would have handled eleven variables. "After prompting ChatGPT 6.0, we were led to the improved bound dim I ⩽ ⌊(4n − 3)/3⌋ in Lemma 3.8", using a Galois-descent approach (Lemma 3.6) that ChatGPT 6.0 suggested. They also used ChatGPT 5.5, 5.6, 6.0 and 6.1 to combine estimates across parameter ranges and for proofreading, and Codex to produce a Lean auto-formalisation of the main theorem that still assumes five results from the literature. They thank OpenAI's Mark Sellke for Pro-model access given in May 2026.
As of 7 October the paper is an unrefereed preprint. No reactions from number theorists had been found yet; it appeared on the same day as OpenAI's 722-manuscript release, which drew most of the attention.
Why it matters
This is one of the oldest benchmark problems of the circle method, and a well-known group of analytic number theorists has now claimed it in full, at the sharp number of variables. It shows a pattern that differs from OpenAI's autonomous releases: a long human project in which a frontier model supplied one step the authors say they could not otherwise reach. For a model with an older cutoff, the record for cubic forms is no longer 14 variables.
Changelog
- 2026-10-07: created (sweep 2026-10-07, arXiv AI-disclosure section; PDF AI statement read)
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Sources (2)
- paperarXiv 2610.08226: The Artin–Davenport conjecture on cubic forms
- codeGitHub: wangyangvictor/cubic-lean-mid (Codex Lean auto-formalisation)
id: 2026-10-06-artin-davenport-cubic-forms-ten-variables · updated 2026-10-07 · open in the interactive timeline