Fermat's Last Theorem/Historical Note

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Historical Note on Fermat's Last Theorem

Many of Fermat's theorems were stated, mostly without proof, in the margin of his copy of Bachet's translation of Diophantus's Arithmetica.

In $1670$, his son Samuel published an edition of this, complete with Fermat's marginal notes.

Fermat's Note

As Fermat himself put it, sometime around $1637$:

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

Loosely translated from the Latin, that means:

The equation $x^n + y^n = z^n$ has no integral solutions when $n > 2$. I have discovered a perfectly marvellous proof, but this margin is not big enough to hold it.

Nobody managed to find such a proof, until it was finally proved by Andrew Wiles in $1994$.

It is seriously doubted that Fermat actually had found a general proof of it.

It is almost impossible that he found Wiles' proof, since it uses areas of mathematics that were not yet invented in Fermat's time.

Fermat himself left an outline of the proof for $n = 4$.

Euler provided the full proof for $n = 4$ and also a proof for the more difficult case $n = 3$.

The cases for $n = 5$ and $n = 7$ were proved by Gauss, Legendre and Dirichlet and others.

The case for $n = 7$ was proved by Gabriel Léon Jean Baptiste Lamé in $1840$.

At the time Wiles' proof was published, the theorem had been established as true for all numbers up to $125 \, 000$.

Furthermore, where $n$ does not divide any of $x$, $y$ or $z$, it had been proved for all $n$ up to $253 \, 747 \, 889$.

In the words of David Hilbert:

Before beginning I would have to put in three years of intensive study, and I haven't that much time to waste on a probable failure.

Donald E. Knuth, in the introductions to early editions of his The Art of Computer Programming, introduced Fermat's Last Theorem as an example of a $50$ point exercise, that is, a research problem.

In later editions, that is, those published after $1994$, it was downgraded to a $45$ point question.