My question to any mathematician reading this: does the above make ANY sense to you?
I ask that because I can read most technical material related to computer engineering, programming, hardware specifications etc. Even if I don't fully understand all details, I can follow them pretty well. So I wonder if professional mathematicians can look at the above and still make sense of it like experienced software engineers do for computer stuff.
But not an expert on this.
While I don't know the specifics, and someone more "in-the-field" than me would recognize all the "named" theorems etc
I am aware that there have been minor issues that have come up with the formalization specifically, and that previous proofs for lower values of n were always needed.
Though it used to be n=5 and lower needed to be checked.
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...
Ugh we still don't know if this is true and it's nearly impossible to calculate without a full understanding of the real CAPEX cycle. Stop spreading these rumors until we know for sure.
It's like having new solar panels installed every week. Sure you're "profitable" on the $0.20/kWh you're selling your "free" energy at when you ignore the cost of the solar panels you're buying every week.
^ 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.
How can you be so sure its not result of inefficiency?
I have used Fable for formalisation and it will, unless I catch it, reprove results it previously had proven, inline, in other results.
> 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.)
You probably heard about Goedel Incompleteness -- the proof that the the axiomatic itself cannot be proven, like using ZFC to prove ZFC, but that's another topic.
It would be fun to play with this Anthropic/Lean formalization under different axiomatics.
Godel theorems are for systems with basic arithmetic, zfc doesn't include arithmetic, thus are not subject of Godel theorems.
zfc itself is not sufficient, you need some layers of extra concepts formalization to fit specific problem domain(e.g. zfc doesn't define even basic arithmetics), which also could have potential issues.
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:
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 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.
Child mortality is very low now compared to the past, thanks to the modern medicine and technology.
I am glad humanity "played God", and reduced this unnecessary child suffering.
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?
I'd also say people may want more life for themselves, but what does that mean at scale, forever?
There are many reasons that people living forever would be a problem for society, the most obvious being an ever-increasing population.
My experience is that it takes a lot of human input to make Fable write code nice enough for a formalisation library others can work on. But since this is certainly a lot of prerequisites formalised as well, it would be nice if not all of the effort was wasted on one capstone proof!
So in the end, it required tooling crafted by humans.
I’ll also share a Python package I wrote for automated theorem proving that has been super useful in my own research [2].
> 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.
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.
i don't really understand either take. nothing else in the world is so perfectly black or white. there will be good, there will be bad.
i think i especially dislike the "100% strictly negative" take, considering the good things that ai has already done or accelerated.
- 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
I'm just old enough to remember Paul Erdo"s and his notion of 'The Book', which he defined to be a book the "Supreme Fascist" (God) had which held the most elegant proofs of mathematical theorems.
https://en.wikipedia.org/wiki/Paul_Erdős#Personal_life
It would be interesting to see how Erdo"s would name such a huge proof by Claude using Lean.
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?
https://leodemoura.github.io/blog/2026-8-1-postmortem-for-ke...
Note to other users: don’t downvote this kind of comment, answer it.
encode mathematical reasoning in a way that can’t be fooled.Notably, junk theorems are true. Nobody would debate that the junk theorem is true. The main thing people would say is that junk theorems, while being true, are sensitive to precisely how you encoded mathematics, so despite being true, they are perhaps not conceptually meaningful.
As an example of a junk theorem, sasy you use the definition of the natural numbers using von neumann ordinals
https://en.wikipedia.org/wiki/Set-theoretic_definition_of_na...
Then for any natural numbers n, m, they're implicitly sets. So n \intersect m = min(n,m). This is the wrong way to think about natural numbers. You should not use this ever in proofs. But this isn't because your proofs would be false, but instead because it is a fundamentally confusing way to think about the natural numbers. It is in this sense it is a "junk theorem".
False negative = could not find a proof of a true theorem.
False positive = erroneous proof of a theorem.
If its 13 million LoC, it might involve so much spaghetti that its unusable other than the result
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
What is even the point? Have claude do it.
I'm not trying to be snarky here. I'm being serious. What is the point? This is an important question that needs to be answered. If something is definitively better, why not have that something take over?
I know people talk about the importance of human endeavor or the "joy" of doing something. But I don't care for those answers because it's weak. The question is deeper than this. AI is better than us, what is the logical point other than attempting to monopolize human effort even though it is inferior.
This was initially "completed" in the 80s. You can see the timeline for cleaning up the proof in e.g. this mathoverflow answer
https://mathoverflow.net/questions/114943/where-are-the-seco...
it's something that some people have been waiting decades for, and is not yet completed.
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.
I hope soon enough we will have one of the big ones proved by AI!
status: "self-assessed"
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.
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'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!
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:
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?