Hacker News Reader

Comment by timmg | Hacker News Reader

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?

View on HN

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.

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.

More likely he decided to leave it in as a nerdsnipe rather than he wrote it in the first place as a nerdsnipe (seems more likely he thought he had it?)

Or he forgot about it? Why should he have a margin note at the top of his mind?

I make notes all the time that I accidentally discover years later with some amusement.

Yea I’m partly saying he came across it or remembered it much later and was amused to not correct it

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...

Among well-known mathematicians, Gabriel Lamé claimed a proof in 1847 that was assuming unique factorization in cyclotomic fields.

This was not obvious at the time, and in fact, Ernst Kummer had discovered the assumption to be false some years before (unbeknownst to Lamé) and laid down foundations of algebraic number theory to investigate the issue.

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).

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

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.

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.

The proof is pretty trivial anyway so it’s left as an exercise to the reader

    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

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).

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)

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.

why does that make it more likely?

Because if he had the general proof he wouldn't need to go out of his way to prove n=4, since it would be covered already by the general proof

It is simply an obvious fault line in the nature of the problem statement: you can crack the problem in 2 parts: the x^4+y^4=z^4 part, and the part that claims x^p+y^p=z^p with p a prime.

Suppose Fermat solved the proof by using this natural fault line -its just how this cookie crumbles- solved the n=4 case, and then smashed his head a thousand times against the problem and finally found the prime n proof.

He challenges the community, and since they don't take up the challenge, "encourages" them in a manner that may be described as trollish, by showing how to do the n=4 case. (knowing full well the prime power case proof looks totally different)

That's an interesting take but I think it's unlikely for two reasons:

1. In any case you view it, it's not trivial, which was the statement in the note. If it were, the effort to publish just for n=4 would be silly, because it would take equal effort to just publish for general case. That he withheld the proof just to mess with people is highly unlikely.

2. I definitely do not make private notes in my books just so that maybe somebody later on would pick up that book and wonder whether I had indeed discovered the secrets of the universe. I definitely do not write "challenges to the community" there.

In what note by whom was it stated that FLT is trivial?

https://mathworld.wolfram.com/FermatsLastTheorem.html

That URL does not substantiate any claim by anyone that FLT was trivial to prove.

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.

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 :)

> we have an upper bound

Is Wiles' proof even in ZFC?

I would be surprised if it wasn’t. Maybe some part of depends on the continuum hypothesis, but ZFC is pretty powerful

Your question is explored in https://www.cs.umd.edu/users/gasarch/BLOGPAPERS/fltlargecard...

Thanks. So if I read this correctly - there is consensus that Wiles' proof can be reduced to ZFC and PA (and maybe even much weaker theories). But as presented Wiles proof relies on Grothendieck's works and Grothendieck could not care less about foundationalism, so no such reduction is known and we don't really have a lower bound even for ZFC.

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.

> 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."

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.

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

Don't confuse majority for consensus, soon the majority will flip, but the consensus will stay the same: that there is no consensus.

P.S. This is utter nonsense:

"Don't confuse majority for consensus, soon the majority will flip, but the consensus will stay the same: that there is no consensus."

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

No support for symbols, amirite?

Its a "dog ate my homework" situation

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.

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.

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.

Euclid’s Elements “rigorous proof” is not the same thing as the modern rigorous proof at all.

>But the infinitesimal methods used before epsilon-delta have been redeemed by the work on nonstandard analysis.

This doesn’t mean that these infinitesimal methods were used in a rigorous way.

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

Reply on news.ycombinator.com