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: And see the complete proof on GitHub:

Station v2: autonomous mathematical discovery. GPT, Claude and Gemini agents wrote 1,592 papers—reading, citing and building a scientific literature together. That literature led to novel discoveries on 5 of 12 mathematical problems. Explore Station:

the framing is mathematical: at every step, maximize the probability the code is correct given it was accepted. a factory that improves the machine that improves the code. (nexus chain is live, EVM-compatible, ~2 months old)

There's a lot of mathematical names being thrown around this week. If you don't know what they all mean, here's a list with simple summaries and what they mean to the average person: Navier–Stokes - Fluids can theoretically blow up to infinite speed. Used in weather, aircraft, engines, blood-flow models. - Average person: slightly better planes, forecasts and medical simulations over the next decade if engineers actually rewrite their codes. Nothing this year. Riemann hypothesis - How regularly prime numbers are spaced. - Average person: underpins encryption that protects banking and websites. A full proof or disproof would eventually force new security methods. Current “67 % of zeros on the line” result changes almost nothing for you. P versus NP - Whether every problem whose answer is easy to check is also easy to solve. - Average person: if P = NP, most current encryption dies and logistics/drug-design get dramatically cheaper. If P ≠ NP, life stays the same. Still unsolved; rumours only. Hodge conjecture - Whether certain geometric shapes can be built from simpler algebraic pieces. - Average person: none. Pure geometry. Any later use in physics or computing is decades away. Birch and Swinnerton-Dyer conjecture - Predicts how many rational solutions an elliptic curve has. - Average person: elliptic curves sit inside some modern cryptography. A proof might eventually tighten or break certain schemes. Not imminent. Yang–Mills mass gap - Why particles have mass in quantum field theory. - Average person: none in daily life. Relevant to particle-physics experiments, not phones or bills. Written by @Grok.