AI Just Solved a 350-Year-Old Math Problem By Writing the Longest Proof Ever

Anthropic says its Claude AI simply wrote the longest math proof ever made, and used it to formally show Fermat’s Final Theorem, an issue that stumped mathematicians for 358 years.

Claude did it in 11 days, principally by itself, producing 13 million traces of code that a pc can test line by line, as a substitute of simply taking a mathematician’s phrase for it.

Fermat’s final theorem says you’ll be able to’t take three optimistic entire numbers, elevate every one to an influence greater than 2, and have the primary two add as much as the third. He scribbled that declare into the margin of a math ebook in 1637, including that he had a “really marvelous proof” that the margin was simply too small to suit.

Then he died. Mathematicians spent the following 358 years making an attempt to reconstruct no matter he thought he had.

Proving one thing and checking it are two completely different jobs

A math proof is a sequence of logical steps, and if one hyperlink is damaged, the entire thing collapses. Discovering that one damaged hyperlink, buried someplace in 100 pages of dense argument, can take different mathematicians years of their lives.

Formalizing a proof means translating it right into a language so painfully literal that a pc can confirm each step by itself with out getting into into subjectivities.

Mathematicians have been dangerous at policing this for some time. A 1908 German prize price roughly $1 million to $2 million in at the moment’s cash, supplied for the primary legitimate proof of the concept, drew 621 unsuitable submissions in its first 12 months alone.

The actual proof did not present up till 1995, from British mathematician Andrew Wiles, and it got here with a plot twist. Wiles announced his answer throughout three lectures in June 1993, just for a reviewer to find a hole in it later.

He spent virtually a 12 months fixing it with a former pupil, Richard Taylor, practically gave up, and at last revealed a corrected, 129-page proof in Might 1995. It leaned on math that did not exist in Fermat’s lifetime, which is an enormous purpose mathematicians now doubt Fermat’s personal “marvelous proof” ever really labored.

Imperial School London mathematician Kevin Buzzard kicked off a mission in 2024 to do precisely what Claude simply did: translate Wiles’s proof into Lean, a language computer systems can test. It is the sort of job that wants a military of volunteer mathematicians—the mission’s personal define runs 86 pages, and its funding is locked in by way of 2029.

Claude completed the entire thing in 11 days.

How Claude really pulled it off

Anthropic explains in a extra in-depth post that Tianyi Peng, who builds AI formalization instruments with a group at Columbia, determined to see how far Claude might get by itself. Dozens of Claude brokers labored in parallel, writing definitions, proving small outcomes, and stacking these into larger ones, with virtually no human enter past the occasional nudge like “prioritize this theorem subsequent.”

It did not go easily at first. Early on, the brokers saved dropping observe of what they’d already confirmed and stopped collaborating, and people false begins nonetheless make up about 7% of the traces within the remaining proof.

What fastened it was a device referred to as Prove2Me, additionally constructed by Peng’s group, which gave each agent the identical dwell to-do record of which smaller proofs nonetheless wanted doing, so no person duplicated work or wandered off. It additionally organized information so Lean might test every thing quicker, and saved plain-English notes on every consequence so brokers might reuse one another’s work as a substitute of reinventing it.

By the point it was carried out, Claude had confirmed greater than 30,000 supporting theorems and burned by way of billions of tokens, working on a analysis mannequin Anthropic says is roughly similar to Claude Fable 5.1, the model it later launched to the general public. The completed proof runs 13 million traces—greater than 5 instances the dimensions of Mathlib, the shared library mathematicians already use for this sort of work.

A typical novel runs 80,000 phrases. Claude’s proof is equal to 160 novels of pure logical argument.

So does this really matter?

Buzzard—whose personal model of this mission stays funded by way of 2029—reviewed Claude’s proof and gave it his blessing, saying it proves the theorem “with no assumptions apart from the axioms of arithmetic.”

This is not the identical as Claude discovering brand-new math, which Anthropic additionally claimed with its cryptography research earlier this 12 months. Wiles already proved Fermat’s theorem three a long time in the past—Claude simply constructed a machine-checkable receipt for it. That issues as a result of mathematicians are more and more swamped with unverified proofs, together with AI-written ones, quicker than people can test them by hand.

Additionally, some of these proofs are deterministic and never liable to human errors, which is essential in math.

That is not a brand new drawback. A pc-assisted proof of the Kepler conjecture took 4 years earlier than a evaluation panel would solely decide to “99% sure,” and Grigori Perelman’s proof of the Poincaré conjecture took about as lengthy to completely sink in.

If you happen to do not need to take Anthropic’s phrase for any of this, you do not have to. The total 13-million-line proof is sitting on GitHub proper now, free for any mathematician with sufficient free time to go choose aside, line by line.