Math: Solving hard problems and discovering new mathematics

In 2000, the Clay Mathematics Institute named seven of the hardest unsolved problems in math the “Millennium Prize Problems” and offered $1 million for a solution to each. Before 2026, only one had ever been solved.

One of them is about the Navier-Stokes equations, the math that describes how fluids like water and air move. Engineers use these equations to design airplanes, pipelines, and weather forecasts. Because the equations are too complicated to solve exactly, engineers usually approximate them on supercomputers. The prize question asks whether a smooth flow of fluid always stays smooth, or whether it can suddenly “blow up,” with its speed shooting to infinity at a single point in a limited amount of time. The question dates back about 90 years.

On September 8, 2026, OpenAI announced that an unreleased internal AI model had answered the question: yes, a fluid can blow up. OpenAI described the model as much more capable than its GPT-6 Astra model. The effort started September 1. About 10,000 copies of the AI, called agents, worked in groups for 88 hours. They sent each other 2.7 million messages and wrote about 130 billion tokens (pieces of words) of text. OpenAI gave separate groups different goals. Some tried to prove that fluids always stay smooth, and others tried to prove that they can break down. The group that succeeded produced a 166-page proof, plus a version written in Lean, a computer language that checks every step of a proof automatically. OpenAI said the effort cost millions of dollars.

The proof covers the version of the problem that allows a smooth outside force to push on the fluid, which the official prize rules permit. It shows that a fluid starting at rest can, with a carefully designed push, reach infinite speed in a limited time while its total energy stays limited. Whether a fluid with no outside push can blow up is still an open question.

The result came with a dispute over credit. About 12 hours before OpenAI’s announcement, mathematician Tristan Buckmaster of New York University announced related results. He had worked on them for about a year with Levent Alpöge, a mathematician employed by the rival AI company Anthropic, as a personal project. Both built on an approach from Spanish mathematicians Diego Córdoba and Luis Martínez-Zoroa. Buckmaster said news of their progress had leaked to OpenAI and may have shaped its effort. OpenAI said it began its push after hearing rumors that Anthropic had solved Millennium Prize Problems. It later said an investigation confirmed that Buckmaster’s use of its products could not have influenced its system. Later analysis reported by Scientific American found that OpenAI’s proof builds its key ingredient in a completely different way from the Spanish approach.

On September 11, the Clay Mathematics Institute said the problem had apparently been settled. However, it only awards prizes after a solution is published in a peer-reviewed journal and examined by the math community, and it said its review would be deliberately slow. OpenAI said it will not claim the prize. Even top experts have found the proof hard to read. Two mathematicians, Zhen Lei and Xiao Ren, have started publishing a more readable version of part of it and called OpenAI’s work a major advance.

On October 6, 2026, OpenAI posted 722 math papers written by an unreleased internal AI model to a public page on GitHub, a website for sharing code and files. OpenAI says the papers cover 372 open problems in math and theoretical computer science, either solving them or making major progress on them. OpenAI linked the model to the one behind the Navier-Stokes proof. It was pointed at about 4,000 open problems, mostly with a single instruction to solve each one. On average, each result it kept used about as much computing as three hours of thinking by ChatGPT’s most powerful paid tier.

The claimed results include a proof of the Unique Games Conjecture, a central question in computer science about which problems can be solved approximately, which researchers had worked toward for over two decades. Another paper describes a faster method, in theory, for multiplying large grids of numbers called matrices. It lowers the key measure of speed to 2.25, from a previous record of about 2.37. Matrix multiplication makes up a large share of the world’s computing, including AI, but the new method is not yet faster in practice. Mathematician Steven Strogatz compared the jump to a record-shattering long jump. Other papers describe a way to multiply huge whole numbers slightly faster than a limit many experts thought could not be beaten, and a solution to the Kakeya conjecture in four dimensions (see Dueling proofs). Several papers disprove long-standing conjectures by finding counterexamples.

One paper made progress on the Riemann hypothesis, one of math’s most famous unsolved problems, which is about how prime numbers are spread out. The paper does not prove the full hypothesis, but it proves a weaker version that is strong enough for many uses. Mathematician Alex Kontorovich wrote, “If a human did this, it would be an instant Fields Medal,” referring to math’s highest honor.

Checking all of this will take time. Only 162 of the 722 papers had their main results checked by Lean, a computer program that verifies proofs step by step. Within a day, OpenAI withdrew three papers after a sign error broke one proof and two others that depended on it. It also revised 14 papers to repair their proofs. The release did not include a full solution to any of the remaining Millennium Prize Problems. As of mid-October, mathematician Terence Tao said he had reviewed about 10 of the papers, and the full set had not been checked.

Reactions were mixed. Tao said many of the proofs seemed to introduce clever new ideas. He also said he was frustrated that no human authors were available to answer questions, give talks, or train students, and he worried that years of patient work by graduate students on these problems had been disrupted. One group of mathematicians called the release a show of power rather than scholarship. Others started building on the results right away. Within days, people using publicly available AI coding tools had improved the number-multiplication result further. The day before OpenAI’s release, Anthropic published a result found by its Claude AI and written up by two computer scientists it hired. It gave the first real speedups for two basic computer science problems whose standard methods were widely believed to be the best possible.

On October 26, 2026, OpenAI announced three more results from its internal AI models, in math, computer science, and physics.

The first is about the Kakeya conjecture, which asks how little space you need to turn a needle so that it points in every direction. It sounds like a puzzle, but it is central to harmonic analysis, the study of how mathematical objects can be built out of waves. Progress had been slow, one dimension at a time. A closely related conjecture about waves, called local smoothing, was proposed by Christopher Sogge in 1991 and proved in two dimensions in 2019. Hong Wang and Joshua Zahl proved the Kakeya conjecture in three dimensions in 2025, one of the biggest math results of the century so far, and Wang won a Fields Medal in 2026. OpenAI says its model proved local smoothing in every dimension at once, which also settles the Kakeya conjecture in every dimension. The proof uses a new helper equation that no one had tried before. OpenAI says the model produced the proof on October 18 and a Lean-checked version the next day.

The second result is about the Exponential Time Hypothesis (ETH), proposed in 1999. ETH says that a famous logic puzzle called 3-SAT cannot be solved much faster than by checking an exponentially growing number of possibilities. Most experts treated it as a bedrock rule of computer science. A search that Anthropic had its Claude AI run found more than 800 research papers with results that depend on ETH being true. OpenAI’s model found a method for 3-SAT that is slightly faster than ETH allows, which means ETH is false. OpenAI says the model found it on October 16 and checked it in Lean the next day. Hours after OpenAI’s announcement, Anthropic posted its own disproof of ETH. Anthropic said an internal Claude research model found the key method on its own, also on October 16, writing about 14 million tokens with no human input. Russell Impagliazzo and Ramamohan Paturi of the University of California, San Diego, who first proposed ETH, are now leading an independent check of OpenAI’s result with Ryan Williams of the Massachusetts Institute of Technology.

The third result is in physics. Another Millennium Prize Problem asks whether the math behind the Standard Model of particle physics, called Yang-Mills theory, is sound. It has two parts, called “existence” and “mass gap.” OpenAI claimed to prove the existence part. The mass gap part remains open. OpenAI has not released this proof. Mathematician Ethan Sussman of Northwestern University, working with Anthropic’s Levent Alpöge, had been using AI to complete an older roadmap toward the same proof and believed they were close. The Advisory Group on Mathematics and Artificial Intelligence (AGMAI), an expert panel OpenAI formed after criticism from mathematicians, has asked OpenAI to release the paper. OpenAI’s model, its instructions to the model, and the amount of computing it used remain private. OpenAI says it is not seeking academic credit.

A major weakness of AI math has been reliability. AI can write proofs that look right but contain small errors, so experts have to check them carefully. In May 2026, researchers at Google DeepMind described AlphaProof Nexus, a system that pairs Google’s Gemini 3.1 Pro AI model with Lean, a program that checks every logical step of a proof. The AI writes a proof attempt, Lean points out errors, the AI fixes them, and the cycle repeats until the proof passes or the AI runs out of attempts.

The researchers ran it on 353 open problems posed by the famous mathematician Paul Erdős and his collaborators. The system solved 9 of them on its own, including two that had been open for 56 years, at a cost of a few hundred dollars per problem. It also proved 44 of 492 open conjectures from the Online Encyclopedia of Integer Sequences (OEIS), a huge online catalog of number patterns. Working alongside mathematicians, it settled a question in algebraic geometry that had been open for about 15 years and helped with research in graph theory, number theory, and quantum optics (the physics of light). In optimization, the math behind many engineering and machine learning methods, it found new settings for an algorithm and proved that the algorithm reaches its answer faster than had been shown before.

Lean also caught the AI’s mistakes. When the AI failed, it often hid the hardest step inside a placeholder, or claimed that a step was a known result from published research when no such result existed. Lean refused to accept these shortcuts.

A simpler version of the system, which just looped between the AI and Lean, also solved all nine Erdős problems, though it cost more on the hardest ones. Versions using smaller AI models solved none. The authors noted that most Erdős problems were still out of reach for their system.

The International Mathematical Olympiad (IMO) is the world’s most prestigious math competition for students under 20. Over two days, contestants try to solve six extremely hard problems, each worth 7 points, for a top score of 42. In July 2026, 666 students competed in Shanghai, and only seven earned a perfect score.

After the human contest ended, AI companies were given the same problems. Two Chinese tech companies, Huawei and Xiaohongshu (known in English as RedNote), said their AI models earned perfect scores under the IMO’s official grading process. Xiaohongshu said no human help was allowed during testing.

Separately, Deedy Das, an investor at the firm Menlo Ventures, tested models that the public can use. He gave the six problems to each model in a simple setup with no internet access and no hints, then had other AI agents grade the proofs strictly. Three models scored 42 out of 42: Anthropic’s Claude Fable 5, OpenAI’s GPT-5.6 Sol, and Kimi K3 from China’s Moonshot AI, an open model anyone can download. Claude Fable 5 was the only one to score perfectly on its first try, at a computing cost of about $51 for all six problems. The other two reached perfect scores after being shown a reviewer’s list of flaws and trying again. Other models and settings scored between 13 and 37. Das’s runs were graded by AI, not by human judges. Anthropic also reported that its Claude Opus 5 model earned a perfect score, with human experts grading one solution per problem.

This was a big step up from earlier years. In 2024, Google DeepMind’s AI reached silver-medal level by solving four of six problems over two to three days. In 2025, AI from Google and OpenAI reached gold-medal level with 35 points, but five students outscored them with perfect scores. After his 2026 test, Das wrote, “The frontier of AI has officially moved well past IMO math.”