Project to formalise a proof of Fermat’s Last Theorem in the Lean theorem prover

https://imperialcollegelondon.github.io/FLT/

by ljlolel • 11 months ago

146 107 11 months ago

imperialcollegelondon.github.io

racl101• 11 months ago

These days Fermat would say: "I have an elegant proof but I don't wanna learn LaTex just to publish it."

generationP• 11 months ago

And soon it will be "but I don't want to learn Lean".

rck• 11 months ago

This is the Lean blueprint for the project, which is a human-readable "plan" more or less. The actual Lean proof is ongoing, and will probably take a few more years. Still cool though.

cubefox• 11 months ago

At the time of writing, these notes do not contain anywhere near a proof of FLT, or even a sketch proof. Over the next few years, we will be building parts of the argument, following a strategy constructed by Taylor, taking into account Buzzard’s comments on what would be easy or hard to do in Lean.

So the title of the paper is misleading at this time.

kevinbuzzard• 11 months ago

That is correct, the title is currently misleading (arguably the title of every paper I ever wrote was misleading before I finished the work, I guess, and the work linked to above is unfinished). If you are interested in seeing more details of the proof I'll be following, they are here https://web.stanford.edu/~dkim04/automorphy-lifting/ . This is a Stanford course Taylor gave this year on a "2025 proof of FLT".

golol• 11 months ago

Hey Mr. Buzzard I want to say I find your work and enthusiasm with Lean and formalization very cool.

eig• 11 months ago

Slightly misleading title- this is the overall blueprint for a large ongoing effort by the Imperial College London to formalize FLT in Lean, not the proof itself (which is huge).

The project webpage has more information about the efforts and how to contribute:

https://imperialcollegelondon.github.io/FLT/

aardvark179• 11 months ago

Thank you. I saw the headline and was thinking things had progress surprisingly quickly.

ljlolel4• 11 months ago

That’s the title I saw!

tomhow• 11 months ago

We've updated the title and URL now, thanks!

timmg• 11 months ago

So: as I understand it, Fermet claimed there was an elegant proof. The proof we've found later is very complex.

Is the consensus that he never had the proof (he was wrong or was joking) -- or that it's possible we just never found the one he had?

wk_end• 11 months ago

It's possible we never found the one he had, but it's pretty unlikely given how many brilliant people have beaten their head against this. "Wrong or joking" is much more likely.

umanwizard• 11 months ago

The former.

We can't be 100% certain that Fermat didn't have a proof, but it's very unlikely (someone else would almost surely have found it by now).

Someone• 11 months ago

Unlikely, but not unheard of. Fermat's theorem on sums of two squares is from 1640. https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_... says:

“Fermat usually did not write down proofs of his claims, and he did not provide a proof of this statement. The first proof was found by Euler after much effort and is based on infinite descent. He announced it in two letters to Goldbach, on May 6, 1747 and on April 12, 1749; he published the detailed proof in two articles (between 1752 and 1755)

[…]

Zagier presented a non-constructive one-sentence proof in 1990“

(https://www.quora.com/What-s-the-closest-thing-to-magic-that... shows that proof was a bit dense, but experts in the field will be able to fill in the details in that proof)

prerok4• 11 months ago

Well, true, we cannot be 100% certain, but if he published the proof to n=4, we can say it's most likely he did not find the general proof.

ape4• 11 months ago

Its a "dog ate my homework" situation

jfengel• 11 months ago

Fermat lived for nearly three decades after writing that note about the marvelous proof. It's not as if he never got a chance to write it down. So it sure wasn't his "last theorem" -- later ones include proving the specific case of n=4.

There are many invalid proofs of the theorem, some of whose flaws are not at all obvious. It is practically certain that Fermat had one of those in mind when he scrawled his note. He realized that and abandoned it, never mentioning it again (or correcting the note he scrawled in the margin).

griffzhowl3• 11 months ago

It was called Fermat's last theorem because it was the only one of the theorems stated by Fermat that remained to be proved at the time

mdiesel• 11 months ago

He probably did know it, it's remarkably simple. I can't remember how to format maths in a HN comment though to put it here.

prerok4• 11 months ago

Yeah, I just figured out how to simply reconcile general relativity and quantum mechanics, but I am writing on my phone and it's too tedius to write here.

throw46395• 11 months ago

    import FLT 
    theorem PNat.pow_add_pow_ne_pow
        (x y z : ℕ+)
        (n : ℕ) (hn : n > 2) :
        x^n + y^n ≠ z^n := PNat.pow_add_pow_ne_pow_of_FermatLastTheorem FLT.Wiles_Taylor_Wiles x y z n hn

jerf• 11 months ago

There is known to be a number of superficially compelling proofs of the theorem that are incorrect. It has been conjectured that the reason why we don't have Fermat's proof anywhere is that between him writing the margin note and some hypothetical later recording of the supposed proof, he realized his simple proof was incorrect. And of course, saw no reason to "correct the historical record" for a simple margin annotation. This seems especially likely to me in light of the fact he published a proof for the case where n = 4, which means he had time to chew on the matter.

jacquesm• 11 months ago

Or, maybe he had a sense of humor, and made his margin annotation knowing full well that this would cause a lot of headscratching. It may well be the first recorded version of nerdsniping.

RainyDayTmrw• 11 months ago

What are some believable but wrong proofs of FLT? Wikipedia also claims that there were historically a lot of them[1], but doesn't provide examples.

[1] https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem#Prizes...

hokkos• 11 months ago

I also have an elegant proof, but it does't quite fit in a HN comment.

glitchc• 11 months ago

No support for symbols, amirite?

Xcelerate• 11 months ago

I feel like there’s an interesting follow-up problem which is: what’s the shortest possible proof of FLT in ZFC (or perhaps even a weaker theory like PA or EFA since it’s a Π^0_1 sentence)?

Would love to know whether (in principle obviously) the shortest proof of FLT actually could fit in a notebook margin. Since we have an upper bound, only a finite number of proof candidates to check to find the lower bound :)

tliltocatl2• 11 months ago

we have an upper bound

Is Wiles' proof even in ZFC?

bubblyworld• 11 months ago

Even super simple results like uniqueness of prime factorisation can pages of foundational mathematics to formalise rigorously. The Principia Mathematica famously takes entire chapters to talk about natural numbers (although it's not ZFC, to be fair). For that reason I think it's pretty unlikely.

zarzavat• 11 months ago

Fermat was alive in the 1600s, long before the advent of mathematical rigour. What counted as a proof in those days was really more of a vibe check.

DoctorOetker7• 11 months ago

It is possible to discover mathematical relation haphazardly, in the style of a numerologist, just by noticing patterns, there are gradations of rigor.

One could argue, being a lawyer put Fermat in the more rigorous bracket of contemporary mathematicians at least.

ants_everywhere• 11 months ago

This is actually way false. Rigorous mathematical proof goes back to at least 300 BCE with Euclid's elements.

Fermat lived before the synthesis of calculus. People often talk about the period between the initial synthesis of calculus (around the time Fermat died) and the arrival of epsilon-delta proofs (around 200 years later) as being a kind of rigor gap in calculus.

But the infinitesimal methods used before epsilon-delta have been redeemed by the work on nonstandard analysis. And you occasionally hear other stories that can often be attributed to older mathematicians using a different definition of limit or integral etc than we typically use.

There were some periods and schools where rigor was taken more seriously than others, but the 1600s definitely do not predate the existence of mathematical rigor.

monkeyelite8• 11 months ago

Not true. Even if it’s more strict it’s just a matter of inserting more care and steps, not changing the original idea.

DoctorOetker7• 11 months ago

The consensus is that there is no consensus yet.

I possess a very simple proof of FLT, and indeed it does not fit in a margin.

I don't ask you to believe me, I just ask you to be patient.

jibal6• 11 months ago

No, the consensus among mathematicians is that Fermat did not have a proof.

Lucasoato• 11 months ago

In the words of mathematical historian Howard Eves, "Fermat's Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published."

iconjack• 11 months ago

From the introduction: "At the time of writing, these notes do not contain anywhere near a proof of FLT, or even a sketch proof."

esafak• 11 months ago

Would writing a Lean proof enable algorithmically searching for a shorter (simpler) proof?

agentcoops2• 11 months ago

I worked a long time ago on tools for algorithmically pruning proofs in another theorem-proving environment, Isabelle. That project was largely just using pretty straightforward graph algorithms, but I’m sure the state-of-the-art is much more advanced today (not least it’s an area where I assume LLMs would be helpful conjoined with formal methods).

lgas3• 11 months ago

I'm not an expert but I'd say based on my experience with Lean so far, the answer is yes. eg. I have entered some proofs from the Lean guide and then replaced tactics with `apply?` which searches for tactics to meet the goal, and eg. reduced this example from the guide:

    theorem and_commutative (p q : Prop) : p ∧ q → q ∧ p :=
      fun hpq : p ∧ q =>
      have hp : p := And.left hpq
      have hq : q := And.right hpq
      show q ∧ p from And.intro hq hp

To this:

    theorem and_commutative' (p q : Prop) : p ∧ q → q ∧ p := by
       exact fun a ↦ id (And.symm a)

Presumably the same thing could be done for each part of a more complicated proof and could be done so at each step automatically.

frakt0x90• 11 months ago

I know nothing about theorem provers. Is the idea that you can prove a large number of simpler statements that build on each other until it all implies the top level theorem you're proving (like how IRL math works)? That way you don't have to understand the gargantuan computer program all at once?

It just seems like it would be as hard to verify the accuracy of the code written to prove a complex theorem like FLT as a manuscript written in English. But if you can rely on smaller statements that build on each other, it would make more sense.

agentcoops2• 11 months ago

More or less! Just like when writing a typical program or indeed mathematical proof, you have a lot of freedom in how you break up the problem into subproblems or theorems (and rely on “libraries” as results already proven). Traditionally the problem with these theorem-proving environments was that the bulk of even relatively basic theorems, let alone the state-of-the-art, had yet to be formalized, so you weren’t quite starting from the axioms of set theory every time but close enough. Lean has finally seen enough traction that formalizing highly non-trivial results starts to be possible.

sestep• 11 months ago

If I understand correctly, the only parts you'd need to verify are the core of the theorem prover, called the "kernel" (which is intentionally kept very small), and possibly the code in charge of translating the theorem statement into the language of that core (or you could look directly at the translated theorem statement, which for FLT I'd imagine may be smaller than the code to translate it). Those pieces are orders of magnitude smaller than the proof itself.

codebje2• 11 months ago

You're basically right, with one teensy caveat: you can't prove the kernel is correct. No model can prove that it is itself consistent, and if you use a more powerful model to prove consistency you're just shifting the goalposts - an inconsistent powerful model can show any model to be consistent.

The kernels are kept small so that we can reasonably convince ourselves that they're consistent.

The expression of FLT in Lean isn't complicated, because FLT isn't complicated; the kernel language of Lean isn't all that far from the surface language; and the expression "∀ a b c n : ℕ, n ≥ 3 → a ≠ 0 → b ≠ 0 → c ≠ 0 → a^n + b^n ≠ c^n" only needs the infix operators converted to lambda applications (though "only" does a lot of lifting here!)

cocoto• 11 months ago

The way it works is that if you trust the theorem prover to be bug free, then you only have to verify that the theorem was correctly stated in the program. The proof itself becomes irrelevant and there is no need to verify it.

dmytrish• 11 months ago

Humans still have to state the goal and write a proof of it, but the proof is computer-verified. It's not irrelevant, except in the sense that any two different ways to prove the same statement are equivalently valid proofs.

drdeca• 11 months ago

1: yes, it is split into proofs of many different propositions, with these building on each-other.

2: The purpose of this is largely for the “writing it” part, not to make checking it easier. The computer checks the validity of the proof. (Though a person of course has to check that the formal statement of the result shown, is the statement people wanted to show.)

ww520• 11 months ago

From the little I know of, language based provers like Lean allow mathematicians to specify the steps of a proof in a precise language format that can be “run” by a machine assisted prover to prove its validity. It automates the verification and makes proofs reproducible.

In the past when proofs were done by hands, little mistakes or little changes could lead to a complete failure of the proof. Mathematicians spent weeks or months to redo the steps and recheck every little detail.

Machine assisted prover basically raises the abstraction level for theorem proving. People don’t need sweat about little details and little changes in the proof steps.

Language based machine provers also enable wider corroboration as a problem can be subdivided into smaller problems and farmed out to different people to tackle, perhaps to different experts for different parts of the problem. Since everyone uses the same language, the machine can verify each part and the overall proof once the parts come together.

crystal_revenge• 11 months ago

At least for theorem proovers like Lean, they're basically type checkers but for much more sophisticated type systems than you find in your average programming language. Basically, if the proof compiles, it is proven. In fact, if you look at Lean code it is very similar to Haskell (and the book Functional Programming with Lean is a great way to explore the language from that lens [0])

You can also think of this in reverse and it might help you understand better: Your type checker at compile time is basically providing a sort of "proof" that all the functions and arguments in your program are consistent. Of course, because the type system is not as sophisticated, it can't prove your program is correct, but it as least can prove something about how your program will behave. If you had a more advanced type system, you could, in fact, prove more sophisticated things about the performance of your code (for example that the shapes of all your matrices matched).

A great book on this topic is Type Theory and Formal Proof [1].

0. https://lean-lang.org/functional_programming_in_lean/

1. https://www.cambridge.org/core/books/type-theory-and-formal-...

openasocket2• 11 months ago

You can think of the formal proofs that mathematicians write as essentially pseudocode, as compared to Lean which is actual code that can be compiled. As an example, it would be common to say “2*x + 4 = 0, therefore x is -2”, in a mathematical proof. But in a computer-readable proof, you have to actually go through the algebraic manipulations, citing what you are doing and what theorem/lemma/rule you are applying. Modern systems like Lean make that much easier, but that’s the essential difference.

Advanced proofs essentially just consist of a series of assertions “X, therefore Y, therefore Z, …” and the justification for that isn’t always laid out explicitly. As a result, when you read a formal proof, it often takes some work to “convince yourself” the proof is valid. And if a proof has a mistake, it’s probably not is one of those assertions, but rather in how you get from assertion X to assertion Y. It can often be really subtle.

Disclaimer: I have a math undergraduate, and have played around with the theorem proving language Coq, but haven’t worked with Lean

voidmain• 11 months ago

If you want to get an idea of what proving things in Lean is like you could try this super fun game, which I think is by one of the people behind the FLT formalization project:

https://adam.math.hhu.de/#/g/leanprover-community/nng4

moi2388• 11 months ago

If you’re interested, here are some really fun games/puzzles to learn the basics of Lean: https://adam.math.hhu.de/

thdhhghgbhy2• 11 months ago

A big concern is how future proof this Lean code is, and they are going to produce a lot of it.

lgas3• 11 months ago

I am assuming you mean it's a bigger concern in Lean than other languages because of what people are attempting to do with it (formalize all of math)? Because the language itself, somewhat by design and somewhat by nature, seems to be more future proof than any other language I've ever used.

thdhhghgbhy2• 11 months ago

So if this project was completed in Lean 3, for sake of example, how compatible would it be with Lean 4 today?

majikaja3• 11 months ago

I think an important thing for something like Lean to gain traction is the availability of learning resources (math 'textbooks') based on the technology so people actually use it for day to day mathematics instead of just formalizing what has already been proven without it.

I don't think many people are going to read Rudin (etc.) then try to rewrite the book or do the exercises in Lean by themselves or read through the mathlib files to see how everything is defined/proved in full generality.

Like a lot of people (I imagine), I'm not a professional mathematician working on advanced, complex problems but I would like to have the ability to have the computer check my solutions to exercises and maybe even aid me with hints for tackling problems if needed.

If math textbooks gave free use for people to rewrite them into computer format and post online it would make life a lot easier.

joachimma• 11 months ago

Terrence Tao is apparently in the process of converting the exercises from his "Analysis I" book into Lean.

https://terrytao.wordpress.com/2025/05/31/a-lean-companion-t...

majikaja3• 11 months ago

This is cool

I guess intros based on the structure of mathlib could work for people who haven't published their own textbooks.

griffzhowl3• 11 months ago

Have you tried "Mathematics in Lean"? It's in the Books section here, along with a few other resources:

https://leanprover-community.github.io/learn.html

majikaja3• 11 months ago

Yes I am aware of this one, it seems like a good intro.

I am just thinking more along the lines of - other than times where a computer is not available to us, why do we keep using traditional notion/pen and paper at all?

If everything is digitalized no one will have mark homework ever again.

monkeyelite8• 11 months ago

I still don’t understand the excitement here other than it’s related to computers. Will formalizing it make it easier for us to understand the core ideas or arguments? I don’t think it will - that would best be done by reading Wile’s paper which is written with that goal in mind.

Given that it was a research frontier where arguments assume an educated audience, it's probably very difficult to formalize.

kobebrookskC3• 11 months ago

the proof is so complicated it's hard to trust if it's only on paper. if we formalize it, it can be checked by a computer.

monkeyelite8• 11 months ago

y/n of strict logical checking the least interesting part of a proof. It's the insight into why it's true or false that's valuable, and that insight of Wiles was enough to convince the rest of the math community.

In other words, the chance that we find gaps and mistakes in the written proof? 100% - the chance we find out it's false due to sloppy logic? 0%.

kevinventullo2• 11 months ago

I contend it is the only way to move forward on the goal of “automating” mathematics. Although we’ve seen natural language approaches do well at IMO, the human effort required to verify higher level proofs is too great with hallucinations being what they are. With something like Lean, you don’t need a human verifier.

monkeyelite8• 11 months ago

With something like Lean, you don’t need a human verifier.

The purpose of a proof is to show yourself and someone else why something is true. I don’t know what it would mean to be writing them for computers to verify. Unless the only thing you are interested in is y/n

LegionMammal978• 11 months ago

With something like Lean, you don't need a human verifier.

Human verification can never be ruled out entirely with these sorts of systems: you always have to check that the definitions used in the final statement mean what you think they mean, and that all of the base axioms are acceptable.

And of course, there's always the possibility of bugs in the kernel. I even recently found a bug [0] in a verifier for Metamath, which is designed to be so simple that its only built-in logic is typed string substitution. But such bugs should hopefully be unlikely in non-adversarial settings.

[0] https://github.com/metamath/metamath-exe/issues/184

jibal6• 11 months ago

https://github.com/ImperialCollegeLondon/FLT/blob/main/GENER...

"It will not be the original Wiles/Taylor-Wiles proof, but rather a "newer" proof which takes into account more recent developments due to Khare-Wintenberger, Kisin and many other people."

ookdatnog• 11 months ago

I watched this video years ago and found it interesting. It's a talk by Kevin Buzzard, a pure mathematician who got really interested in theorem proving software, and he explains his motivation.

https://youtu.be/Dp-mQ3HxgDE?si=8a0d6ci-7a-yfhou

griffzhowl3• 11 months ago

He's one of the leaders of this project to formalise Fermat's last theorem too

monkeyelite8• 11 months ago

Thanks

PeterStuer• 11 months ago

You mean you never tried to prove Fermat's in your head when going to bed in the evening? Works great as a sleeping aid.

triknomeister• 11 months ago

Mathematics is actually the easiest thing to formalize. It's just the standards of formalization are much higher.

lgas3• 11 months ago

Isn't mathematics the only thing that it's possible to formalize? Doesn't the process of formalizing something else turn it into mathematics?

gtsnexp• 11 months ago

Question for the organizers: Is there an inactivity timeout on claimed sub-problems? Say someone “claims” a task and then goes quiet, others hold back; but without claims, you risk duplicate effort, right? How do you balance that? If there’s a policy already, what counts as “active”? A short status note, a link to a draft, or partial results? If not, would you consider a lightweight scheme:

Soft claim (one task per person); progress update within 7–14 days (longer for big items); auto-release if no update; extensions on request; Tag some tasks as “pairable” so two people can collaborate openly.

Curious how you’re handling throughput vs inclusion here.

jesuslop• 11 months ago

Section #7 is on the Langlands Conjectures. "Conjetures in a proof blueprint?" Curious about the role in the proof.

"The { Wiles } modularity theorem { that cracked FLT } is a special case of more general conjectures due to Robert Langlands. The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field. Most cases of these extended conjectures have not yet been proved. (https://tinyurl.com/yc6kth2m)"

New math is emerging "as we speak" from the group of Gaitsgory (https://tinyurl.com/9r5bsufj), but apply to the (differential) geometric Langlands area, while the quote looks about the algebraic-geometry area (of Weil's triple Rosetta Stone). Yet areas have well understood connections. Both kind of works (new math and formalizing) are very needed. Verifiable proofs have to help to avoid math building collapsing under own weight/richness.

Project to formalise a proof of Fermat’s Last Theorem in the Lean theorem prover

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