Fermat's Last Theorem

Pierre de Fermat

Fermat's Last Theorem was conjectured by the French mathematician Pierre de Fermat. He said he had proved this problem but that there was not enough room in the margin to state his proof:

"Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, 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."

The English translation of Fermat's Latin statement is:

"It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."

This problem has confounded mathematicians for centuries, and there still is no proof for it using elementary techniques. Gauss and other mathematicians doubt that Fermat was able to prove it himself, but Cal Tech mathematics Professor E.T. Bell, who wrote the standard biography of all the great mathematicians, wryly observed that "the fox who could not get at the grapes declared they were sour." "And so for all of [Fermat's] positive assertions with the one exception of the seemingly simple one which he made in his Last Theorem and which mathematicians, struggling for nearly 300 years, have been unable to prove: whenever Fermat asserted that he had proved anything, the statement, with the one exception noted, has subsequently been proved. Both his scrupulously honest character and his unrivalled penetration as an arithmetician substantiate the claim made for him by some, but not by all, that he knew what he was talking about when he asserted that he possessed a proof of his theorem."

The theorem is as follows:

Andrew Wiles

For integers n > 2, there are no nonzero integral solutions to: xⁿ + yⁿ = zⁿ

In the summer of 1986, Ken Ribet proved that Fermat's Last Theorem is a special case of the Taniyama–Shimura Conjecture.

In a series of lectures in 1993, mathematician Andrew Wiles announced a proof using techniques in algebraic geometry, relying on the disfavored Axiom of Choice. A flaw was found before publication. Wiles spent a year trying to fix the flaw, and in September 1994, he and Richard Taylor announced a new version of the proof that is not widely understood but is no longer criticized by university-based mathematicians. However, criticism does continue on the internet.