Yes—AI can prove some theorems by producing a proof in a formal system such as Lean, where a proof assistant checks that the proof satisfies a precisely stated goal. That confirms the derivation relative to the formal statement and assumptions; it does not, by itself, show that the statement faithfully captures the original question. Demonstrations so far are meaningful but bounded: they do not show that AI can prove arbitrary mathematics without human guidance or formalization.
What does it mean for AI to prove a theorem?
The phrase can refer to several different tasks, and they do not provide the same evidence of correctness:
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- Generating an informal proof: an AI writes mathematical reasoning in ordinary language. It may be useful, but fluent prose is not a correctness certificate; intermediate steps can be plausible and wrong.
- Formalizing a problem: someone translates the intended question and its assumptions into a precise proposition in a formal language. This is a separate task from proving it, and a translation can encode the wrong statement.
- Searching for a formal proof: an AI looks for a derivation of that proposition in a proof assistant such as Lean. The assistant checks whether the resulting formal artifact satisfies the encoded goal.
- Helping discover mathematics: a model identifies patterns, examples, or conjectures that guide mathematical work. This can contribute to discovery without itself completing a formal proof.
The most precise claim is therefore that a system produced a proof in a named formal system and that the proof checked—not simply that it “proved mathematics.”
What has AI actually proved?
The 2024 International Mathematical Olympiad
Google DeepMind reported that AlphaProof and AlphaGeometry 2 together solved four of the six problems at the 2024 International Mathematical Olympiad (IMO), earning 28 of 42 points, within the silver-medal range under the contest’s scoring. AlphaProof solved two algebra problems and one number-theory problem; AlphaGeometry 2 solved the geometry problem. The two combinatorics problems remained unsolved. The contest problems had first been manually translated into formal mathematical language, so this result did not show that the systems independently converted the original statements into formal goals. DeepMind reported that one solution took minutes and others up to three days. Google DeepMind’s 2024 IMO report describes the benchmark and its results.
#1 Best Overall
Formal proofs accepted into Metamath
In 2020, OpenAI reported that GPT-f found short proofs accepted into the main Metamath library. This is a historical example of machine-generated proof work with a formal proof system, not evidence of what any current system can do on arbitrary mathematical problems. OpenAI’s GPT-f report describes that work.
Results and Lean formalizations announced in 2026
On October 6, 2026, OpenAI announced that it was releasing mathematical results and Lean formalizations for many proofs, along with details about how results were obtained and compute estimates. OpenAI said it 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. The announcement also said the organization continues to work on citation quality, exposition, and presentation. A release announcement is not the same as independent peer review, and it should not be read as evidence that every result has been formally verified. OpenAI’s October 6, 2026 announcement sets out its claims and qualifications.
What does Lean check—and what does it leave open?
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; Lean checks whether that derivation satisfies the proposition encoded in the system. A successful check is strong evidence that the derivation is valid under the formal setup.
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But there are two distinct questions: whether the proof follows from the formal statement, and whether that formal statement accurately expresses the problem a person meant. Lean checks the first. It cannot alone guarantee the second, confirm that the chosen assumptions match the intended question, or supply an explanation that makes the proof’s significance clear to a reader. In the IMO demonstration, the statements were manually translated before proof search, making that human formalization step especially important. Microsoft Research describes Lean as a functional programming language and interactive theorem prover.
Rank #3
Can AI help discover new mathematics?
Yes, though discovery assistance is not identical to a machine-checked proof. A 2021 Nature paper described machine-learning methods that helped mathematicians identify patterns and develop contributions connected to an open problem in topology and a candidate algorithm in representation theory. The work was interactive: computational pattern recognition informed mathematicians, who interpreted and developed the findings. It is best understood as machine-learning-assisted mathematical discovery, not as a chatbot independently producing and verifying the results. The 2021 Nature study describes the approach.
Why are current demonstrations limited?
A checked proof can be rigorous without showing general mathematical competence. The main limitations are different problems, and should not be collapsed into one claim:
- Formalization: translating a natural-language problem and its assumptions into the right formal proposition can require mathematical expertise. A flawless proof of a mistranslated proposition does not answer the original question.
- Coverage: success on a contest, a formal library, or a particular mathematical domain shows performance within that scope; it does not establish universal ability.
- Search and reasoning: systems can fail to find a proof that exists, and informal natural-language reasoning can contain plausible but incorrect steps. DeepMind has noted limitations in reasoning and training data as obstacles to general mathematical problem solving.
- Explanation and context: formal validity does not by itself explain why a result matters or provide exposition a mathematician can readily assess. OpenAI’s 2026 announcement itself identified citations, exposition, and presentation as areas for improvement; it did not report a measure of reader understanding.
For that reason, a benchmark result should be read with its task count, domain, formalization process, time allowance, and availability of a checkable artifact—not as a general ranking of mathematical intelligence.
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Use these questions to distinguish a verified result from a promising-sounding demonstration:
- What did the system output? Was it informal prose, a conjecture, a formal statement, or a machine-checkable proof?
- Who formalized the problem? Did the system translate the original question, or did people supply a prepared formal statement? Were its assumptions checked against the intended problem?
- What checked the proof? Identify the proof assistant or checker, and see whether the proof artifact is available for inspection.
- What was the scope? Look for the benchmark or mathematical domain, number of tasks, and known failures. A contest result and a discovery study measure different activities.
- How much human guidance and resource use were involved? Check for disclosed interaction, search time, and compute, keeping organization-reported estimates attributed to their source.
- What is the mathematical contribution? Establish whether the result is a benchmark solution, a shorter proof, a useful conjecture, or a new result—and whether its assumptions and context are explained.
These checks keep a formal proof, an AI-assisted conjecture, and an impressive contest score from being treated as interchangeable evidence.
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