The Gaussian moat problem
A Gaussian prime is an irreducible element of , viewed as a point of the plane. For a step bound , join two Gaussian primes when their distance is at most . Large prime-free disks exist with centers on any line (Gethner, Wagon, Wick), but a walk in the plane can go around them, and computations bound only the component of a particular starting region (Tsuchimura, steps up to 6 from the origin). Gethner and Stark (1997) formulated a version uniform in the starting point and proved it for step bounds and by periodic sieving. Is it impossible, for every finite , to walk to infinity through distinct Gaussian primes with steps of length at most ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Gaussian primes and sieve methods
- Posed by
- Basil Gordon, at the 1962 International Congress of Mathematicians in Stockholm (per Gethner and Stark, Periodic Gaussian moats, Experimental Math. 6, 1997); Erdos credited Gordon and Motzkin
- Year posed
- 1962
- Years open
- 64y
- Solved
- 2026-09-26
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 40 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for every real there is a finite such that every connected component of the graph on Gaussian primes with edges of length at most has at most vertices, independent of the starting prime and including primes on the axes; so every walk through distinct Gaussian primes with steps at most has at most terms. The proof finds, for each , a finite set of split primes whose joint sieve leaves a periodic set with no infinite bounded-step walk, using geometric sampling and entropy estimates. is nonexplicit, so no numerical moat width is given for any particular .
What the AI did
The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorems 1.1 and 1.2 of the TeX source; the proof was not refereed. Lean-checked on the Comparator challenge GaussianMoat, listed in the release's formalization catalogue (OAI.GaussianMoat.fullMain, OAI/NumberTheory/GaussianMoat/Main.lean). Its statement has two parts: for every real there is no injective sequence of irreducible Gaussian integers with successive distances at most ; and for every real one natural number bounds the size of every connected component of the graph joining distinct Gaussian primes at distance at most , and the length of every injective bounded-step walk. That is the headline claim, axis primes and associates included. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.