Yes: AI can prove some theorems by producing a proof in a formal system such as Lean, where a proof assistant checks the proof against a precisely encoded statement. That verifies the derivation as formalized; it does not by itself show that the encoded statement faithfully represents the original question. AI has also helped mathematicians find patterns and conjectures, a related but distinct kind of mathematical assistance. Current results are significant but bounded: they do not show that AI can prove arbitrary mathematics without guidance or formalization.
What does it mean for AI to prove a theorem?
The phrase can describe several different tasks. Keeping them separate makes claims about AI and mathematics easier to evaluate:
- Writing an informal proof: A model produces mathematical prose. The explanation may be helpful, but fluent text alone is not a certificate of correctness; intermediate steps can be plausible and wrong.
- Formalizing a problem: A person or system translates a natural-language question and its assumptions into a precise proposition in a formal language. This translation can misstate the intended problem.
- Searching for a formal proof: A system tries to construct a proof of that encoded proposition in a proof assistant such as Lean.
- Checking a proof: The proof assistant verifies whether the formal proof artifact follows the rules of the system and establishes the encoded proposition.
- Helping discover mathematics: A model or machine-learning method identifies patterns, examples, or conjectures that may guide mathematicians, without necessarily producing a complete formal proof.
For precision, say that a system “produced a Lean proof that checked” when that is what happened. This identifies the artifact and the checker instead of implying that a model’s prose has been independently validated.
What has AI actually proved?
AlphaProof and AlphaGeometry 2 at the 2024 IMO
Google DeepMind reported that AlphaProof, a reinforcement-learning-based system for proving statements in Lean, and AlphaGeometry 2, a geometry-solving system, together solved four of the six problems at the 2024 International Mathematical Olympiad (IMO). They received 28 of 42 points, which DeepMind said was within the silver-medal range. AlphaProof solved two algebra problems and one number-theory problem; AlphaGeometry 2 solved the geometry problem. The two combinatorics problems were not solved. DeepMind’s account of the IMO result describes the systems and outcome.
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The contest statements were manually translated into formal mathematical language before the systems worked on them. DeepMind reported that one solution took minutes and others took up to three days. The result is evidence of substantial performance on a specific, prestigious contest—not proof that AI can solve any research problem or independently interpret arbitrary written mathematics.
GPT-f and the Metamath library
In 2020, OpenAI reported that GPT-f found short proofs that were accepted into the main Metamath library. This is an earlier example of AI-assisted formal proof generation, not a measure of what current systems can do. OpenAI’s GPT-f report describes the work.
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OpenAI’s October 2026 mathematics release
On October 6, 2026, OpenAI announced mathematical results and Lean formalizations for many proofs, along with information about how the results were obtained and compute estimates. OpenAI said it continues to work on the quality of citations, exposition, and presentation. Its announcement is an organization’s release, not independent peer review, and the announcement alone does not establish that every result has been formally verified. OpenAI estimated roughly three hours of ChatGPT Pro thinking-equivalent compute per average result; that is the company’s own estimate, not an independently measured benchmark. OpenAI’s announcement is the source for those release claims.
What does Lean verify—and what does it not?
Lean is a functional programming language and interactive theorem prover used for formal mathematics. A formal proof is a derivation represented in a precise language; a proof assistant checks that artifact against a formal proposition using the system’s rules. Microsoft Research describes Lean as “a functional programming language and interactive theorem prover.” Lean’s official project site provides an entry point to the ecosystem.
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A successful check is strong evidence that the derivation establishes the proposition that was encoded. It does not, by itself, establish that:
- the encoded proposition is a faithful translation of the original informal question;
- the assumptions in the formal statement are the assumptions the reader intended;
- the result is new, important, or relevant to a broader mathematical problem; or
- the formal proof’s presentation gives a human reader the insight needed to understand why the result matters.
That is why the manual translation of the 2024 IMO problems matters. The translation stage and the proof-search-and-checking stage are distinct. A proof assistant checks the latter relative to the formal statement; human mathematical review may still be needed for the former.
Can AI discover new mathematics?
AI-assisted discovery is broader than completing a proof. A 2021 Nature paper describes machine-learning methods that helped mathematicians identify patterns and develop contributions related to an open problem in topology and a candidate algorithm associated with representation theory. The work is presented as an interaction: machine learning helps surface patterns, and mathematicians interpret them and develop the mathematics. It is therefore more accurate to describe this as machine-learning-assisted discovery than to claim that a chatbot independently wrote and verified the results. The 2021 Nature paper details the work.
What are the limits of current AI theorem proving?
DeepMind’s account of the IMO result says current AI systems still struggle with general mathematical problems because of limits in reasoning and training data. It also cautions that natural-language systems can hallucinate plausible but incorrect intermediate reasoning. A benchmark result should therefore be read with its scope and method attached, not as a general measure of mathematical ability.
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- Formalization: Translating a problem and its assumptions into a correct formal statement can require mathematical expertise. The IMO statements were manually translated.
- Coverage: Performance on a particular contest, topic, or formal library does not establish broad capability across mathematics.
- Search and reasoning: A system may fail to find a proof even when one exists, or produce invalid reasoning in informal prose.
- Human understanding: A checked artifact establishes formal validity relative to its assumptions, but explanation and context still matter to people reading the result.
Microsoft Research’s undated Lean project page says the community’s formalized mathematics exceeded one million lines of code and described coverage of over half of the undergraduate mathematics curriculum. These are figures reported on an undated project page, not a dated, independently measured assessment of AI capability. They describe the formal mathematics ecosystem, not the range of theorems an AI can prove. The Lean project site is the source for that project-page description.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How should you compare claims about AI math systems?
When two demonstrations sound similar, check whether they actually performed the same task. These questions expose the important differences:
- What did the system output? Informal prose, a conjecture, a formal statement, or a machine-checkable proof?
- Who formalized the problem? Did the AI translate the original question, or did people supply a prepared formal statement?
- What checked the proof? Is a proof assistant named, and can the formal artifact be inspected?
- What was the scope? Which benchmark or mathematical domain, how many tasks, and what known failures?
- What interaction and resources were involved? Was there human guidance, how long did search take, and were compute estimates disclosed?
- What mathematical contribution was made? Was it a known contest solution, a shorter proof, a useful conjecture, or a new result explained with its assumptions and context?
For example, the IMO report gives a contest, a score, manual formalization, and reported solution times. The 2021 Nature work concerns discovery assistance rather than the same proof-search task. Treating those as a single leaderboard would obscure what each result demonstrates.
How can a computer check a proof?
In a proof assistant, a proof is encoded as a formal object, and the checker determines whether it satisfies the rules for proving the target proposition. To explore machine-checked mathematics, readers can start with Lean’s official project and learning resources. The Lean project site links to information about the language and its ecosystem; Microsoft Research also describes university courses and supporting literature as part of that ecosystem.
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