Kaplansky's zero-divisor conjecture
For a group and a field , the group algebra consists of finite formal sums of elements of . If has finite order then , so torsion forces zero divisors. The domain conclusion is known for unique-product groups, torsion-free elementary amenable groups and several geometric classes. Kaplansky's zero-divisor conjecture asserts the converse. If is torsion-free and is a field, is free of nonzero zero divisors?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Group rings and geometric group theory
- Posed by
- Graham Higman (1940 thesis) and Irving Kaplansky (1956 conference, published 1957, Problem 6)
- Year posed
- 1940
- Years open
- 86y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 62 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims a counterexample: a finitely presented torsion-free group with a finite two-dimensional classifying space and nonzero with . The group is the fundamental group of a cone complex over two random labelled graphs, with cancellation forced by odd label intersections; torsion-freeness comes from asphericity. It works in characteristic two only and does not disprove the zero-divisor conjecture over or , nor the Atiyah conjecture.
What the AI did
The OpenAI math release (github.com/openai/math) states that its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author. The release includes a Lean 4 formalization of the main theorem.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source against the posed problem, and the Lean statement. formalization.yaml lists OAI.TorsionFreeZeroDivisors.main (file lean/OAI/Algebra/GroupRing/Main.lean, comparator ComparatorChallenges/TorsionFreeZeroDivisors.json, axioms propext, Quot.sound, Classical.choice). The comparator statement asserts a group that is finitely presented, torsion-free in the ordinary sense, has a finite connected 2-dimensional CW complex with contractible covering as , and has nonzero in MonoidAlgebra (ZMod 2) G with . That is the full headline claim. Not rebuilt here; the comparator run was not reproduced.