Comment by timmg

3 days 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?

44 comments

timmg

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jerf  3 days 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  3 days 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.

    • ljlolel  3 days ago

      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?)

      2 replies →

  • RainyDayTmrw  3 days 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...

    • a57721  3 days ago

      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.

jfengel  3 days 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).

  • griffzhowl  3 days 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  3 days 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.

  • prerok  3 days 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.

    • thiht  3 days ago

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

  • throw46395  3 days 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

umanwizard  3 days 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).

wk_end  3 days 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.

Xcelerate  3 days 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 :)

Lucasoato  3 days 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."

DoctorOetker  3 days 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.

  • jibal  3 days ago

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

    • jibal  1 day ago

      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."

zarzavat  3 days 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.

  • ants_everywhere  3 days 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.

    • unexpectedtrap  3 days ago

      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.

  • DoctorOetker  3 days 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.

  • monkeyelite  3 days 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.