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Discussion (157 Comments)Read Original on HackerNews
Provides great context on this accomplishment, what it means but also doesn't mean.
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.
Gives you an idea of the scale...
^ this section should have been in the first few paragraphs imho. Explaining why this is relevant shouldn't be so far down.
Explaining the value of what you are showing should always go towards the start. Else, why would anyone bother with the rest?
At $50/M output tokens, this would have cost on the order of $300k (plus a bit for input/prefill tokens) at API rates.
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.
Pretty insane. I suppose it lends further credence to the idea that anything that can be shown to be correct can be done by a model.
My strong hunch is that it was a joke - he knew how difficult the problem was and claiming he had a solution was I think a huge motivating factor for many mathematicians trying to prove it. The greatest nerd snipe troll in history.
> 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.
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.)
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!
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.
And granted, I don't know the exact details about Lean. It might be that they don't have an incredibly simple core - as has elsewise been the norm.
https://leodemoura.github.io/blog/2026-3-16-who-watches-the-...
...and for those who are looking to roll-their-own:
https://ammkrn.github.io/type_checking_in_lean4/title_page.h...
...and some thoughts on putting stuff in the kernel:
https://lawrencecpaulson.github.io/2026/07/30/Collatz.html
If its 13 million LoC, it might involve so much spaghetti that its unusable other than the result
I optimistically expect to witness the advent of a global 'panacea' in my lifetime thanks to AI's efforts. Cost effective large scale genetic engineering, a cure for every disease, potentially even a cure for aging.
The future is both beautiful and terrifying.
If you're happy to die, why be bothered by others' trying to live longer? You won't be around. And assuming people can finance it themselves, is it really a problem for society?
There are many reasons that people living forever would be a problem for society, the most obvious being an ever-increasing population.
I’ll also share a Python package I wrote for automated theorem proving that has been super useful in my own research [2].
[1] https://arxiv.org/abs/2608.25220
[2] https://github.com/henryrobbins/open-atp
Is this basically like opening up a black box and seeing 13 million gears all rotating seemingly randomly and still having no idea how the machine actually works?
[1] https://github.com/ImperialCollegeLondon/FLT
> 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...
> 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.
Hopefully this helps mathematicians. It seems very clear to me that it will help software engineers apply formal methods to more of our software.
- https://www.youtube.com/watch?v=nUN4NDVIfVI (The bridges to Fermat's Last Theorem)
- https://www.youtube.com/watch?v=NPOw4iIxN6o (podcast)
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
Not sure why anyone is excited about this tech.
will you not see that people could be truly empowered and yet will instead be oppressed?
It seems to me that it is making everyone (including myself and the researchers we need to cure diseases) lazy and dependent on thinking machines owned by tech companies. Just how autocomplete and gps made us worse at spelling and navigating, llms make us less able to exercise our ability to think and problem solve. This will have 100% strictly negative consequences on you and the world as a whole. .
And even if there was a cure to many diseases the eugenics types who are embedded in worldwide power structures definately arent going to share that universally.
the project: https://imperialcollegelondon.github.io/FLT/
>>Along the way, it wrote 13 million lines of Lean and proved 29,500 intermediate theorems
Did a human check the 13 million lines of code? How does QA'ing this type of work works?
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.
https://leodemoura.github.io/blog/2026-8-1-postmortem-for-ke...
But how do you know you told it what you intended to tell it?
Note to other users: don’t downvote this kind of comment, answer it.
Wiles’s proof will remain a mystery to me.
Or is it the case that as long as you verify the initial statements you are trying to prove the rest doesn't matter
/s
No we cannot. LLMs do not, by their very nature, understand a single thing. You are giving far too much credence to hype and marketing.
An interesting next target would be formalizing the classification of finite simple groups. The original proof scattered over thousands of pages of journal articles, plus Aschbacher and Smith's 1300 page 2 volume monograph. It's so long it's hard to know if there are any gaps. Researchers have been working on a streamlined new proof, but it's already many volumes long.
I hope soon enough we will have one of the big ones proved by AI!
Fable, please translate to HOL-light. Make no mistakes. You are doing great!
A human mathematician writes a Lean proof:
- Unlikely that the mathematician would cheat with Lean bugs or even know how to find one. Trust increases.
An AI writes a Lean proof:
- AIs have been "ambitious" in their goals in the past and do know how to find Lean bugs and exploit them. Trust decreases.
I've tried the various intros to Lean multiple times (even before Lean 4 came out) and something about the way Lean proofs are written does not align with how I think about proofs. My very brief attempts at Isabelle / RCoq feel more natural.
I think it's a pity that the future of proofs is Lean. I'd love for someone to come up with a more digestable proof language!
Lean is not for humans.
https://news.ycombinator.com/item?id=49203626
It is truly saddening to think that machines will deprive us of this wonder and experience.
But truly exciting to dream about what lies beyond the limits of our biology.
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
That is just how it is.
Seeing it hit across: the work we used to do outdoors, the sleep-wake-dark cycle we adhered to for millennia, and more
Can not we do it by code?