Checking that a major mathematical proof is correct can take years. Formalization—converting the mathematical reasoning into a form computer proof assistants like Lean can verify—can help. Last month, Claude completed the first formalized proof of Fermat’s Last Theorem, one of the most famous theorems of all time. This was a project experts thought would take many years. It is the largest Lean proof ever written. Fermat’s Last Theorem was first proven in 1995 by Sir Andrew Wiles, more than 350 years after it was conjectured. Our proof, which totals over 13 million lines of code, provides machine verification. More importantly, it proves over 29,000 other theorems that the proof requires, across many areas of math which had never before been formalized. We see this as a major step in the long process of firming up the core of mathematical knowledge, building on work from three centuries of mathematicians and hundreds of contributors to Lean and Mathlib. We are optimistic that AI-assisted verification of mathematical proofs will help reduce the burden of refereeing mathematics in an era where more proofs are being produced than ever before. You can read about the process on our Science Blog: anthropic.com/research/forma… And see the complete proof on GitHub: github.com/anthropics/fermat…

Maryam Mirzakhani called herself a “slow” mathematician. She said, “you have to spend some energy and effort to see the beauty of math.” To work through a problem, she spread sheets of paper across the floor, doodled surfaces and shapes, and wrote formulas around the drawings. To her daughter, that was simply “painting.” She never claimed a secret method:“I don’t have any particular recipe [for developing new proofs] … It is like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck, you might find a way out.” That was her genius: not speed, but patience. Not a formula, but staying lost long enough to find a new way through.

How’s this timing: It’s the 1st day of fall semester, my 1st class starts in 15 mins, and this morning my wife went into labor. Logging into Zoom from the hospital to tell my students the news and then dismiss class. Wife is doing amazingly! Maybe 3-5 hours until baby arrives!

Okay, let’s do novels now. Make a novel smaller. I’ll go first: A Tale of One City

Please welcome to the world a beautiful new geometric object, to do with a problem i’ve always loved. claude really contains multitudes:D Does S^6 admit a complex structure? Yup

We asked an unreleased research version of Claude to take a stab at the Riemann hypothesis. It didn’t solve it, but it did make strides on a related problem: it increased the lower bound for the fraction of zeros of the Riemann zeta function that satisfy the hypothesis from 41.6% to 67.2%. anthropic.com/research/riema…

We asked an unreleased research version of Claude to take a stab at the Riemann hypothesis. It didn’t solve it, but it did make strides on a related problem: it increased the lower bound for the fraction of zeros of the Riemann zeta function that satisfy the hypothesis from 41.6% to 67.2%. anthropic.com/research/riema…

Breaking News: Scientists have used A.I. to create new viruses for the first time, raising hopes for medical advances while also raising the possibility that the technology could someday be used to invent dangerous pathogens. nyti.ms/4bxx7Wy

Niels Abel, born exactly 224 years ago today, proved at 21 that there is no algebraic formula to solve equations of degree 5, settling a 250-year-old problem. He paid the printer himself and compressed the proof to 6 pages to cut costs. He sent a copy to Gauss, who never responded. Abel died of tuberculosis at 26. Two days later, a letter arrived announcing he'd been appointed professor in Berlin.