On July 20, Anthropic researcher Levent Alpoge said he used Claude Fable 5 to find a counterexample to the Jacobian conjecture, one of the central open problems in algebraic geometry.
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^3to C^3,…
— levent (@__alpoge__) July 20, 2026
The conjecture was formulated by German mathematician Ott-Heinrich Keller in 1939. In simplified terms: consider a formula that maps one triple of numbers to another. Mathematicians can check “at each point” that, locally, it neither glues together nor loses nearby information. The conjecture claimed that if a formula preserves information locally everywhere, then it is globally invertible — one can recover the input from the output.
Claude Fable 5 produced a formula that violates this. It passes the local check at every point, yet it sends three different input triples to the same output. The original input cannot be recovered, implying the conjecture is false. Alpoge included Wolfram Alpha links where the calculations can be reproduced.
According to Alpoge, a friend asked him about the conjecture. Claude worked on the task during the World Cup final.
Over 87 years, the conjecture became one of the leading unsolved problems in its field. It also became known for incorrect proofs.
Mathematician Qiaochu Yuan said this is the most famous open problem solved by a language model so far. He quoted mathematician T. T. Moh, who in 2008 analyzed a flawed paper. Moh wrote that Beniamino Segre published three incorrect proofs, Claude Chevalley accepted an erroneous proof as correct, and Igor Shafarevich treated the conjecture as a proven theorem.
for some background,
1) the jacobian conjecture is by far the most famous open problem resolved by an LLM so far
2) it’s also infamous for attracting wrong proofs so this is quite funny in a specific way. a comment from a 2008 paper by t.t. moh debunking such a proof:
> The… https://t.co/PymlXIho2P pic.twitter.com/2lwERRGHGP— QC (@QiaochuYuan) July 20, 2026
Mathematician Jared Duker Lichtman called the result “remarkable.” He noted that a special case of the conjecture was the topic of Ethan Zhang’s dissertation — Zhang later made a breakthrough in prime number theory.
This is quite a remarkable result:
Posed in 1939, the Jacobian conjecture is one of the central open problems in algebraic geometry, but was just disproved by Alpoge, Matthew, and Claude Fable 5.
The Jacobian conjecture roughly says that a multivariable polynomial F has an… https://t.co/P9A1ZLocai— Jared Duker Lichtman (@jdlichtman) July 20, 2026
Alpoge’s result has not yet undergone peer review. A key feature is verifiability: AI companies typically cite benchmarks and internal tests, while this counterexample is a concrete formula anyone can plug in and check.
Alpoge earned his PhD at Princeton under Fields Medalist Manjul Bhargava. In 2015, he received the Morgan Prize — the highest U.S. award for undergraduate research in mathematics.
AI and open problems
The counterexample to the Jacobian conjecture is not the first mathematical result obtained with the help of language models.
In October 2025, OpenAI vice president Kevin Weil said GPT-5 solved ten Erdős problems. Hungarian mathematician Paul Erdős was one of the most prolific mathematicians of the 20th century and authored about 1,500 papers. After his death, a collection of hundreds of unsolved problems remained. They are stated briefly and their answers are checkable, so AI developers use the list as a testbed for models.
The claim did not hold up. Soon after publication, it turned out the model had found solutions already present in the literature. After criticism, Weil deleted the post.
In January 2026, GPT-5.2 Pro solved Erdős problem No. 397 on central binomial coefficients. Engineer Neel Somani posed the query to the model. The answer was formalized by Aristotle — an automatic prover from startup Harmonic.
Fields Medalist Terence Tao confirmed the correctness. Earlier, he noted an autonomous solution to problem No. 728, which has no analogs in the literature. At the same time, Tao cautioned that AI is still picking “low-hanging fruit” accessible to standard techniques.
In May, an OpenAI model disproved Erdős’s 1946 unit distance conjecture in the plane. Unlike most items on the list, this conjecture is a famous problem that mathematicians worked on for 80 years. The company called it the first case of this level for AI.
The proof was checked by nine external mathematicians and described in a companion paper. Soon after, Google DeepMind introduced the AlphaProof Nexus system, which resolved nine smaller Erdős problems. Its proofs were verified by Lean.
In July, the Star Fleet Math project presented solutions to 20 Erdős problems obtained by 20 parallel Codex agents. Each proof was verified by the Lean 4 kernel.
In February, Google introduced the AI mathematician Aletheia, which autonomously solved four problems from the Erdős list.
