Thanks! I've added that link to the toptext.

I'd really like to make it the top link (and relegate https://www.anthropic.com/research/formalizing-fermats-last-... to the toptext) since HN has been tracking the work of https://news.ycombinator.com/user?id=kevinbuzzard for a long time and we're big fans. But I guess that would be overkill.

I’m not very good at mathematics, but it seems like Kevin should take his girlfriend on trips more often for the good of all mathematicians.

"I was given £1M to run my project over 5 years; Anthropic took only 11 days but I do wonder if they spent more money…"

Gives you an idea of the scale...

Forgive the authors of the article for assuming readers would complete it.

Nah they should have released it in a 14-part tweet instead.

And human salaries for those who worked on the prover harness etc. which isn't just standard Fable.

It also uses Prove2Me, which uses a graph like previous automated theorem provers. A fact that LLM hawks have categorically denied here before, with opposition naturally flagged.

Now they have it in writing.

But also achievable on a $150/mo (CAD) Max 5 subscription (I currently have 11.6B tokens in the last 30 days) according to /usage. It doesn’t break down input vs. output tokens as far as I can tell.

What would it cost to make a team of mathematicians do the same?

There is no way Fermat could have fit that in the margin. Definitely vindicated.

especially compared to existing 129 pages proof by human

one of the most popular books in india growing up. used to see it everywhere

100% It is a very insightful book

physics is like sex: sure, it may give some practical results, but that's not why we do it

This is a crucial point. There have been many bugs in Lean (and in other proof assistants for that matter). Proof assistants work well on human input, because it was created with a certain intent.

We simply don’t know what those 13M contain and whether it “makes sense” and doesn’t trigger Lean bugs. (There are “independent” lean verifiers, but historically they contained the same, or similar, bugs.)

The AI labs have out considerable effort in trying to find and patch lean exploits. They explicitly set agents and have them try to prove false.

> Daniel used OpenAI internal models to discover new soundness issues in the official Lean kernel and runtime

https://leodemoura.github.io/blog/2026-8-24-postmortem-for-t...

They found several bugs and they have patched them. Lots of work going into making sure lean is sound.

It is possible, although the post notes that the proof was also verified by the Comparator, which means any exploited bug has to also be present in that checker. Which is not unheard of, but is much less likely than merely an exploit in Lean 4.

Not just lean, but math foundation itself, I am not strong expert, but my understanding is that there is no fully recognized axiomatic foundation for modern math, all proposals could lead to some weird results.

Anthropic surely is well aware. Most likely they asked separate agents multiple times to code review the proof and look for exploits.

Nope! :(

Meaning, people and LLMs are finding 1=0 bugs in formal verification tools. I have no idea how likely this is in this case, though!

The proof system is relatively easy to verify.

I am not entirely sure about lean, but the core algebras for systems like lean are in the 100s of lines of code.

You can likely convince yourself it is correct in a weekend or less - especially with an Ai to help you understand it.

It's wild to think that aging is something that needs to be cured, and isn't a part of the natural human experience. I'm so tired of people trying to play the role of God, as well as people that cheer these sorts of things on.

I dont think it will happen. AI models are kneecapped. Only a tiny tiny tiny fraction of people are on the list of even being able to use these tools for such things.

I wonder if AI can come up with mathematical conjectures. As in, they feel it's right but can't prove it. What even happened in Fermat's brain to sense it was true?

The last box, per https://news.ycombinator.com/item?id=49568667

> What this work is, and is not

> I am currently being funded by the EPSRC to formalize a proof of Fermat’s Last Theorem, and a naive reaction to the news above is that I no longer have any work to do. This is not the case. The work certainly achieves some of the aims of the EPSRC project, and indeed it goes much further in terms of what is formalized (I only promised the EPSRC that I would reduce FLT to the 1980s; this repo proves the whole thing). But I also promised several other things to EPSRC: firstly, that I would be making pull requests to Lean’s mathematics library, adding fundamental objects from modern number theory; this is ongoing. And secondly, and perhaps most importantly, that I would be creating a dynamic document enabling humans to explore the modern proof. My guess is that it is unlikely that Anthropic are going to do this; they will feel that their job is done with the formalization (and they did not formalize the modern proof anyway).

> Note that mathematically this work of anthropic tells us essentially nothing: I am on record as saying that I am 99.9% sure that the proof of FLT is OK, and most people in the number theory community are 100% sure (formalization has made me more paranoid about the mathematical literature than most). From my understanding of the argument, the formalization just faithfully follows the early literature on the proof and adds nothing.

https://xenaproject.wordpress.com/2026/09/04/flt-anthropic-h...

Seems to have taken it in good spirit:

> We shared the resulting proof with Kevin Buzzard, who said:

> > This extraordinary autoformalization achievement, which Anthropic researchers say only took 11 days, proves Fermat’s Last Theorem with no assumptions other than the axioms of mathematics. Along the way 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.

This stuck out to me, too. That a (presumably rather simple) coworking tool was instrumental in shaping the vast (6B token!) output is eye-opening. We have this vast power but without intermediate structure it is wasted. Much like Turing machines themselves, which are shaped by language design to get somewhere at the expense of getting everywhere.

And the multiple Numberphile appearances of Ken Ribet are interesting too! He is incredibly well spoken.

- https://www.youtube.com/watch?v=nUN4NDVIfVI (The bridges to Fermat's Last Theorem)

- https://www.youtube.com/watch?v=NPOw4iIxN6o (podcast)

Also recommend his other books!

Big Bang - history of the understanding of space and the universe

Code book - history of the maths of ciphers

Haven’t read them for years but I’ve been meaning to again

You'll get mass poverty and violence which the owners of AI will qwell with AI surveillance and weapons. AI will be used to pit us against eachother and justify wars to keep us busy. Fun times ahead.

Not sure why anyone is excited about this tech.

their reaction here : https://xenaproject.wordpress.com/2026/09/04/flt-anthropic-h...

There is a simple piece of code that can check simple steps, and many people agree this checker is correct. Then there is a formalization of the theorem which many people agree defines the theorem accurately. Then there is 13 million lines of proof that nobody has read, but the proof checker validated each step. That's enough.

So, all you have to verify is the formalization of the theorem, and believe that the proof checker is free of bugs. You don't have to read the actual proof.

This was my question as well. The way I understand it, it's like a compiler, it implements rules, in this case logic/math rules that tell you whether something follows from assumptions you've given it.

But how do you know you told it what you intended to tell it?

A human definitely didn't, but one of the benefits of formal verification is that even if the work done to achieve something is slop-y or excessively verbose, solvers like Lean guarantee that the initial proposition (assuming it was written correctly and in this case was definitely reviewed by humans) is definitively True. This is true across other domains of formal verification outside of math as well

The point of writing Lean code is that Lean checks it accordingly. Lean is a domain specific language to encode mathematical reasoning in a way that can’t be fooled.

Note to other users: don’t downvote this kind of comment, answer it.

The nice thing about theorem provers is that you don't need to read the intermediate lines. You need to make sure that the goal/result actually matches what you think it says - but everything in the middle is validated by the prover.

No. No human checked it. But a type checker did. And that is much better.

Of course it is. The interesting thing is that it was able to produce a Lean proof in 11 days, when there's been an ongoing project for several years to do the same thing (though a somewhat different proof) that is nowhere near done.

About two month ago: https://en.wikipedia.org/wiki/Jacobian_conjecture

That's next week's work.

Lean's proofchecker is a big piece of code, so it's possible that it has a bug (and historically has had some).

Yes. FLT follows from the fact that you can't build the equivalent representation of n-simplex turning into a hypercube in dimensions higher than 2

/s

Yep. There may be only 25-50 people alive today in the whole world who can credibly claim to understand Wiles' proof. Now we add an LLM to that list. Absolutely mind-blowing stuff.

It's a great comedy that we move the buck from "I don't trust the human proof" to "I don't trust the Lean proof" despite the level of trust dramatically increasing. Moving to HOL-light might be another modest increase in trust, but to pretend the implementation of HOL-light has never had bugs and it's kernel could never have a bug is hubris.

The nice thing is, once all of these proofs are formalized in a machine-checkable language, it should be relatively straightforward to translate the corpus between different languages, if someone finds something with a nicer syntax.

If you're doing it for fun anyway, why not use the language that gives you the most pleasure?

Interesting to find this comment, I’ve been dipping my toes into formal methods and was doing a RCoq tutorial yesterday (really basic stuff), and I also noticed that the proofs in RCoq have a more pen -and-paper proof feel to them.

Hearing someone say "the future of proofs is Lean" is a bit like hearing someone say "the future of programming is Rust." Sorry to disappoint, or happy to inform, there are hundreds of programming languages actively being used, and Rust is not even the most used language. To think that proof assistants, fancy programming languages, would be any different is suspiciously motivated.

That's like saying the future of code is Assembler.

Lean is not for humans.

Formalizing is not the same as discovering. There is still plenty of room for human ingenuity.

> It is truly saddening to think that machines will deprive us of this wonder and experience.

It won't deprive us.

Recent video I've watched from Brandon Sanderson, IMO also applies to all the things we love and not just art:

https://youtu.be/mb3uK-_QkOo?si=SG1uvGUbN6SOYI_J

Makes me wonder, if we make a tradeoff for comfort and advancement from our biology's "limits" - and that tradeoff is spiritual fulfillment.

Seeing it hit across: the work we used to do outdoors, the sleep-wake-dark cycle we adhered to for millennia, and more

Just loop through all values of a, b, c, and n?

Sure, go on and try it ;)

Lean _is_ code. FLT cannot be proven by exhaustion because it's domain is an infinite set: the natural numbers above 2.