Fermat's Mysteries

Ariane Coffin remembers Fermat on August 17, his birthday:

Fermat is best known for his little theorem and last theorem. Fermat’s little theorem states that if p is a prime number, then for any integer a, the number a^p − a is an integer multiple of p. He introduced this theorem in 1640 in a letter to a friend, which read:

“Et cette proposition est généralement vraie en toutes progressions et en tous nombres premiers; de quoi je vous envoierois la démonstration, si je n’appréhendois d’être trop long.”
(And this proposition is generally true for all progressions and for all prime numbers; the proof of which I would send to you, if I were not afraid to be too long.)

And on that cliffhanger, Fermat’s little theorem was left unproven until 1683 by Leibniz and again in 1736 by Euler.
As for Fermat’s last theorem, Fermat scribbled it in 1637 in the margin of a book:

“Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.”
(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.)

In other words, no three positive integers a, b, and c can satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than two. Perhaps the most deceptive of all mathematical theorems, Fermat’s last theorem looks simple but quickly became mathematics’ holy grail. Centuries of geniuses unsuccessfully attempted to discover a proof, let alone one elegant enough to be Fermat’s alleged marvelous proof.
Fermat’s last theorem was finally proven by Andrew Wiles, a professor at Oxford University, in 1994 (published in 1995). However, Wiles’ very long and very complex proof used principles of modern mathematics that were completely unknown and unimaginable to Fermat at the time, insinuating that Wiles’ proof was clearly not the same as Fermat’s.
So while Fermat’s last theorem was proven at last, the mystery remains. Is it possible Fermat devised an elegant proof that no one in the world could fathom for centuries thereafter? Did he possess a proof but later realized it was incorrect? Did he outright lie, time after time, about having proofs to his theorems in order to look smarter?

That is really funny.  You have to say that even if he did just lie about having proofs, in fact, he was right.  He probably cranked through enough permutations to convince himself that nobody else was ever going to come up with an example to prove him wrong.  I may have to check out the book, Fermat's Enigma.