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"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." — Fermat (1670), in the margin of his copy of the Arithmetica (translated from Latin) [Wikipedia].

It's remarkable that Fermat never wrote down his proof of what is now his most famous theorem, yet he happened to be right about it. It took hundreds of years and dozens of pages to prove this, yet he was somehow able to conceptualize it on his own while casually reading the Arithmetica. To add, Andrew Wiles' proof of Fermat's Last Theorem recruits a great amount of abstract algebra—a field of math that would not be invented until around 200 years after Fermat made his famous note in that margin.

How was Fermat able to grasp the reasoning behind this theorem's proof without abstract algebra? Is there a simple way to conceptualize Fermat's Last Theorem using only the mathematical knowledge available to Fermat during his time?

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    $\begingroup$ There are lots of popular books and articles about this topic you could read, but to summarise it's considered unlikely that Fermat actually had a proof for all $n$. The proofs for successively increasing values of $n$ are quite accessible, only the general proof by Wiles isn't. $\endgroup$ Oct 21, 2022 at 0:13
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    $\begingroup$ There is certainly no evidence that his proof was correct. Perhaps he tried to write down what he thought was a proof, and then realized it was in error? It's not really a surprising conjecture - he knew about the Pythagorean triples (positive integers $a,b,c$ with $a^2+b^2=c^2.$ It is natural to ask about higher powers, and not finding any, conjecturing that there aren't any. $\endgroup$ Oct 21, 2022 at 0:20
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    $\begingroup$ I think he managed to prove some specific cases (of which only n=4 survives), and mistakenly thought his reasoning generalized $\forall n \ge 3$. $\endgroup$
    – Dan
    Oct 21, 2022 at 0:22
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    $\begingroup$ It is likely that he solved several small cases ($n=4$ for sure and $n=3$ and $n=5$ pretty likely) $\endgroup$
    – Peter
    Oct 21, 2022 at 0:39
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    $\begingroup$ Related (duplicate?): math.stackexchange.com/q/2001462/42969 $\endgroup$
    – Martin R
    Oct 21, 2022 at 7:58

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There is absolutely no way Fermat had a correct proof; he just made a mistake. I don't know if there's hard evidence for this but I was under the impression that the nature of the mistake was something like assuming that the ring of cyclotomic integers $\mathbb{Z}[\zeta_p]$ has unique factorization for all primes $p$, which it doesn't. If it did you could imagine carrying out a proof analogous to the proof for $n = 3$ using the Eisenstein integers $\mathbb{Z}[\zeta_3]$.

Kummer showed in 1847 that FLT for a prime exponent $p$ (and the general case reduces to the case of prime exponent) follows if $\mathbb{Z}[\zeta_p]$ satisfies a property which generalizes having unique factorization involving its class number; this is explained in this note by KConrad. These primes are called regular primes and FLT is (relatively) easy to prove for them. So the hard cases are the "irregular" primes.

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