Direct Consequences of the Three-Dimensional Counterexample to the Jacobian Conjecture

Expository writing following an explicit three-dimensional Keller counterexample.

AI's solution to 87-year-old riddle takes mathematicians by surprise | New Scientist

Mathematicians have been trying to prove the Jacobian conjecture for nearly a century, but now the Claude Fable 5 AI has apparently found a counterexample that disproves it

levent on Twitter / X

hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3,…— levent (@__alpoge__) July 20, 2026

The Jacobian Conjecture is False Per Anthropic (Link in Description)

945 votes, 354 comments. Normally I would be extremely skeptical, but the result is checkable by simple computation. Remarkable! The two-dimensional…

Jacobian Conjecture -- from Wolfram MathWorld

The Jacobian conjecture asserts that a polynomial map F:C^n->C^n having a nonzero constant Jacobian determinant is an automorphism. In the plane, first stated by Keller (1939), it says that a ring map F of C[x,y] (the polynomial ring in two variables over the complex numbers) to itself that fixes C and sends x, y to f, g, respectively, is an automorphism iff the Jacobian f_xg_y-f_yg_x is a nonzero constant. The condition is easily shown to be necessary. There have been at least five...

Terence Tao (@tao@mathstodon.xyz)

In traditional mathematics (or "Math 1.0"), a breakthrough proof of a long-standing open conjecture generates a large amount of subsequent activity and excitement in the field. The authors of the proof are invited to give many talks, and meet with other experts in the area. Workshops are set up to discuss the proof, as well as other recent developments. As a result, new connections and collaborations are formed, followup problems are shared, and other mathematicians (both junior and senior) are attracted to the field (including in some cases the authors themselves, if they initially came from a different subject). As a result of all this activity, the proofs naturally become digested, streamlined, placed in context with other results in the field, and ultimately become part of the textbooks and lecture notes for the next generation of mathematicians in the field. (1/4)

Przemek Chojecki | PC on Twitter / X

The Growing Map of Open Mathematical Problems.We mapped 15,000+ conjectures from UnsolvedMath to show potential links between concepts.It also shows how under formalized the frontier is (less than 10%). pic.twitter.com/nm0PCpXVPf— Przemek Chojecki | PC (@prz_chojecki) August 28, 2026

How I Vibed a Proof of Conway’s Conjecture — overreacted

You can just prove things, apparently.

Amateur armed with ChatGPT 'vibe-maths' a 60-year-old problem

A ChatGPT AI has proved a conjecture with a method no human had thought of. Experts believe it may have further uses

Tony Feng on Twitter / X

I am a mathematician but I avoided commenting on this because the problems are (far) outside my domain. Even if I sat down to read the technical details (which I have not), it would be hard to appreciate the context of prior work, etc. But I've gotten enough sense of things… https://t.co/VZzosBYFto— Tony Feng (@tonylfeng) August 5, 2026

A result does not tell you how it was made - Sensemaker

A correct formula can have an unverified origin, and a successful AI answer can hide a forbidden route. Those claims need different evidence.

Mathematics with large language models as provers and verifiers

During 2024 and 2025 the discussion about the theorem-proving capabilities of large language models started reporting interesting success stories, mostly to do with difficult exercises (such as problems from the International Mathematical Olympiad), but also with conjectures [Feldman & Karbasi, arXiv:2509.18383v1] formulated for the purpose of verifying whether the artificial intelligence could prove it. In this paper we report a theorem proving feat achieved by ChatGPT by using a protocol involving different prover and verifier instances of the gpt-5 model working collaboratively. To make sure that the produced proofs do not suffer from hallucinations, the final proof is formally verified by the lean proof assistant, and the conformance of premises and conclusion of the lean code is verified by a human. Our methodology is by no means complete or exact. It was nonetheless able to solve five out of six 2025 IMO problems, and close about a third of the sixty-six number theory conjectures in [Cohen, Journal of Integer Sequences, 2025].