Anthropic's AI Agents Crack Fermat's Last Theorem Formalization in Just 11 Days

Fermat's last theorem formalised by AI agents in just 11 days | New Scientist

AI company Anthropic has created a formalised proof of Fermat's last theorem. It took just 11 days for a group of AI agents to complete the task, confirming that the human-found proof proposed in the 1990s is correct. Fermat's last theorem puzzled mathematicians for centuries until it was proven in 1995 by Andrew Wiles. It states that there are no whole numbers a, b, and c that satisfy the equation aⁿ + bⁿ = cⁿ, where n is a whole number greater than 2. The theorem though easy to state was fiendishly difficult to prove. Mathematician Pierre de Fermat posed the puzzle in the 17th century and famously alluded to a proof that he claimed to have discovered, saying that it was too large to fit in the margins of the textbook he was writing in. Many mathematicians tried and failed to find a proof. Wiles worked on the problem for seven years in secret before announcing his breakthrough in 1993. Proofs often involve lengthy logical arguments that build on each other. If a single step contains an error then the whole thing can collapse. This happened to Wiles when a flaw was found in his proof that took him and collaborator Richard Taylor around a year to fix. Formalising mathematical theorems is a solution to this problem. It takes them out of the realm of pen and paper, and puts them into a configuration - computer code - that allows machines to grapple with them, methodically working through the logic and exposing any flaws. There are already 2 million lines of formalised mathematics stored in a central repository called Mathlib. Earlier this year a conference of AI experts, computer scientists and mathematicians was held in London by Kevin Buzzard at Imperial College London, who was - until Anthropic's announcement - working on a five-year project to turn Wiles and Taylor's 100 pages of proof into computer code called Lean, so that it can be formally checked for correctness and used as a foundation for further research. Buzzard said at the time that when his project began he was confident of success, but that recent advances in AI had convinced him that the project would be complete much faster than he previously believed possible. Anthropic's announcement has now overtaken that effort and wrapped up the problem. Buzzard said in a statement published by Anthropic that the company's proof leaves "no assumptions other than the axioms of mathematics". In short, the problem is solved. "Along the way we see autoformalisation of algebra, harmonic analysis, geometry and number theory, and we learn that AI autoformalisation artefacts are now robust enough to be built upon; the proof is multi-layered," said Buzzard. "If the automatic formalisation of FLT is possible now, then we have taken a big step towards automatic formalisation of the modern mathematical literature." Anthropic wrote in a blog post announcing the result that its Claude model worked continuously and autonomously for 11 days to write the proof. Numerous separate AI agents were involved, with different tasks like tackling smaller chunks of theorem being assigned to each. Human experts occasionally issued "high-level instructions" to keep the work on track, the company says. And several times the agents "lost track of the project's state and stopped collaborating effectively". Interestingly, the company says that success came after it began using a tool designed for human mathematical collaboration called Prove2Me which helped different agents track their work and decide on next tasks. Anthropic's formalisation runs to 13 million lines of Lean and covers roughly 29,500 intermediate theorems that were necessary stepping stones to completing the overall work. That makes the proof over five times the current size of all previous work on Mathlib - which also makes it the largest Lean proof ever written.

Anthropic uses Claude to formalize proof of Fermat's Last Theorem

Anthropic uses Claude to formalize proof of Fermat's Last Theorem Anthropic PBC has used Claude to create a computer-verifiable version of a famous, highly complicated mathematical proof. The company detailed the project in a blog post published today. A proof is a series of arguments that proves a mathematical hypothesis is correct. The proof that Anthropic tackled verifies a hypothesis called Fermat's Last Theorem. Originally floated in 1637, the hypothesis focuses on the properties of positive whole numbers. The proof of Fermat's Last Theorem was developed in 1995 by mathematician Andrew Wiles. It runs for 129 pages and took months of work to verify. Anthropic's research project formalized Wiles' proof, which means that the company turned it into a form that can be automatically verified by computers. Formalizing proofs is useful because it rules out the possibility of human error and eases information sharing among mathematicians. A formalized proof takes the form of a code snippet written in a programming language called Lean. It's a specialized syntax that mathematicians use to verify hypotheses. Anthropic's proof comprises 13 million lines of Lean code, which makes it the largest-ever file of its kind. Formalization is difficult because proofs tend to be quite terse. They lack certain explanations that a computer would need to understand them, which requires Lean developers to add in the explanations manually. Another source of complexity is that the arguments in a proof often build on one another. That means one erroneous line of Lean code can render all the subsequent code invalid. Mathematicians expected the process of formalizing Wiles' proof to take several years. According to Anthropic, its researchers completed the task in 11 days using an internal research model. The algorithm is described as being roughly on par with Claude Fable 5.1, the immediate predecessor of GPT-6 Astra. Notably, the model completed the task using only a limited amount of high-level input from humans. It spun up several dozen agents that generated six billion tokens of output while working on the proof. Along the way, they proved no fewer than 29,500 intermediate theorems. Anthropic's initial attempt to formalize Wiles' proof was unsuccessful. According to the company, the breakthrough came when it gave Claude access to an open-source tool called Prove2Me. The software makes it easier for AI agents to determine the optimal next step in a lengthy processing workflow. Prove2Me also helps lower inference costs. "We see autoformalization of algebra, harmonic analysis, geometry and number theory, and we learn that AI autoformalization artefacts are now robust enough to be built upon; the proof is multi-layered," said Kevin Buzzard, a mathematician whose work Claude used to generate its formalized proof. The milestone comes a month after Anthropic detailed another LLM-driven mathematical advance. The company used Claude to discover new information about the Riemann zeta function, a closely studied mathematical object. It's the center focus of the Riemann hypothesis, one of the world's most difficult conjectures. Rival OpenAI Group PBC is also harnessing its LLMs to advance mathematics research. Last month, the company used its latest Astra model to solve several Erdos problems and narrow a number of open questions in theoretical computer science.