Artificial intelligence solved 350 years of mathematical problems by writing the longest proof in hi

Anthropic announced: Claude AI wrote the longest mathematical proof in history, formally verifying Fermat's Last Theorem

Anthropic announced that its artificial intelligence model Claude had just written the longest mathematical proof in history, and formalized it proved the "Fermat's Last Theorem" that has plagued mathematicians for 358 years. The entire verification process took only 11 days and was mostly completed independently by AI. Claude generated 13 million lines of code, allowing computers to check logic line by line rather than relying solely on verbal statements from mathematicians.

Fermat's Last Theorem states that there cannot be three positive integers a, b, and c, such that the n-power of a plus the n-power of b equals the n-power of c, where the exponent n is greater than 2. In 1637, Pierre de Fermat wrote this assertion in the margin of a mathematics book and claimed to have discovered a "truly wonderful proof," but the margin was too small to accommodate the details of the proof. Fermat then died, and for the next 358 years, countless mathematicians tried to reconstruct ideas that might exist in his mind.

Proof and Verification: Two very different tasks

Mathematical proof is a series of logical steps. If any link breaks, the entire argument will collapse. Finding the broken link in hundreds of pages of intensive arguments often takes years of effort from other mathematicians. The so-called "formal proof" refers to the transformation of an argument into an extremely rigorous and literal programming language, allowing the computer to independently verify the correctness of each step without involving subjective judgment.

For a long time, mathematicians 'self-regulation in this field has not been perfect. For example, a German prize established in 1908 (approximately US$1 million to US$2 million in today's currency) to reward the first person to effectively prove the theorem received 621 erroneous submissions in its first year.

The real proof didn't appear until 1995, from the British mathematician Andrew Wiles. However, this achievement was accompanied by a dramatic turning point. In June 1993, Wiles published his solution in three lectures, but a reviewer later discovered loopholes. Working with former student Richard Taylor, he spent nearly a year patching the bug, eventually publishing a revised 129-page certificate in May 1995. The proof relied on a mathematical theory that did not exist during Fermat's lifetime, which is the main reason why most mathematicians today suspect that Fermat's so-called "wonderful proof" is actually not true.

How did Claude do it?

Mathematician Kevin Buzzard at Imperial College London launched a project in 2024 to do what Claude had just done: translate Wiles 'proof into Lean, a formalized language that can be checked by computers. This is a task that requires the participation of a large number of volunteer mathematicians-the project's own outline is 86 pages long and funding is locked in until 2029. Claude completed all the work in just 11 days.

Anthropic explained in a more in-depth blog post that Tianyi Peng, who built AI formal tools at the University of Colombia, decided to test Claude's upper limit on his ability under autonomous circumstances. Dozens of Claude agents work in parallel, writing definitions, proving small results, and integrating these results into larger theorems, with little human intervention and only occasionally prompting to "prioritize the next theorem."

Things didn't go well at first. In the early days, agents often got lost in proven content, leading to disruption in collaboration. These trial and error processes account for approximately 7% of the final certification code. The key to solving the problem is a tool called Prove 2Me, also developed by Peng's team. The tool provides each agent with a unified real-time to-do list, listing small certificates that still need to be completed, thereby avoiding duplication of work or deviating from direction. In addition, it optimizes the file structure to speed Lean's inspections and retains English annotations for each result so that agents can reuse each other's work rather than reinventing the wheel.

After the project was completed, Claude proved more than 30,000 support theorems and consumed billions of Tokens. The task was run on a research version model, which Anthropic said was roughly equivalent to the Claude 5.1 version that was later released to the public. The resulting proof was 13 million lines long, more than five times the size of the shared library Mathlib that mathematicians currently use for such work. A typical novel is about 80,000 words, and Claude's proof is equivalent to 160 novels of pure logical argument.

What does this mean?

Bazard reviewed Claude's proof and accepted it, saying that he had proved the theorem "without making any other assumptions other than mathematical axioms." But that doesn't mean Claude has discovered completely new mathematical knowledge-Anthropic made similar claims in his research on cryptography earlier this year. Wiles had proved Fermat's Last Theorem thirty years ago, and Claude had just built a machine-verifiable "receipt" for it.

This is critical because mathematicians are increasingly overwhelmed by unverified proofs, including those written by AI, which are growing far faster than humans 'ability to verify manually. In addition, such proofs are deterministic and are not susceptible to human error, which is extremely important in the field of mathematics.

This is not a new issue. The computer-aided proof of the Kepler conjecture took four years, and the review committee could only promise "99% certainty"; Grigori Perelman's proof of the Poincaré conjecture also went through a long digestion period. If you don't want to believe Anthropic's story, don't have to. The complete 13-million-line proof is now available on GitHub and can be disassembled and inspected line by line by any mathematician with enough spare time.