Fermat's Last Theorem

Fermat's Great Theorem was formulated in the 17th century by Pierre de Fermat, but was not proved until 1994 by Andrew Wiles. The conclusive culmination for the proof is considered to be the collaboration of Wiles with Richard Taylor, which, in addition to the final proof by Wiles, resulted in a simultaneous publication of a partial proof by both Wiles and Taylor as joint authors.

The theorem states: If is a natural number greater than 2, then the n}-th power of no positive integer can be decomposed into the sum of two such powers:

with positive integers is only possible for and

Fermat's Great Theorem is considered unusual in many respects. Its statement, despite the difficulties encountered in its proof, is easily understood even by laymen. It took more than 350 years and was a story of failed attempts, in which, since Leonhard Euler, numerous leading mathematicians such as Ernst Eduard Kummer have participated. Numerous partly romantic, partly dramatic, but also tragic episodes of this story have made it popular far beyond the circle of mathematicians.

The proof finally produced, in the preliminary work of which Gerhard Frey, Jean-Pierre Serre, Barry Mazur and Ken Ribet were also involved in addition to Wiles and Taylor, is regarded as the high point of 20th century mathematics.

Designations

There are different names for this theorem. The most common one in German is Großer Fermatscher Satz and derived from it Großer Fermat in contrast to Kleiner Fermatschen Satz or Kleiner Fermat. Since no proof has been handed down by Fermat himself, strictly speaking it was initially only a conjecture. Therefore the term Fermat's conjecture is also used, but even before the proof was spoken of Fermat's theorem. To include Wiles, the finder of the proof, there is also talk of the Fermat-Wiles theorem. In English, the theorem is called Fermat's Last Theorem, which is sometimes (inaccurately) translated in German as Fermat's letzter Satz or Fermat's letztes Theorem.

Source

Probably between 1637 and 1643, an exact year cannot be given due to the following circumstances, Fermat wrote the following lines as a marginal note in his hand copy of the Arithmetika of Diophantos of Alexandria next to the 8th problem of the second (Greek) "book":

Since Fermat's hand copy of the Arithmetika was found by his son in his father's estate only after his death and he did not date his marginal notes, an exact date cannot be determined. But it is plausible to assume that Fermat had solved at least the case and perhaps also the case before he was tempted to make his equally famous as "careless" remark. Therefore the year of origin is more likely 1641 than 1637.

That Fermat had found a proof for the special case which he perhaps believed to be able to generalize, is obvious, because this special case is an easy conclusion from a theorem explicitly proved by him: Area trianguli rectanguli in numeris non potest quadratus. (The area of a Pythagorean triangle cannot be a square number.), which he wrote including proof in the margin next to the 26th problem of the 6th (Greek) "book" of Arithmetika. André Weil also proved convincingly that Fermat possessed all means to prove also the case with his method.

The theories used in Wiles' proof in 1995 were not even rudimentarily developed over 350 years earlier. This does not exclude with certainty that one day a simpler proof will be found, using more elementary means. But that Fermat could have found one is doubted by most number theorists today. The surest sign that Fermat soon realized that he had not found a proof after all is that he did not mention the theorem and a proof of it to any of his correspondents. Fermat's marginal remark, moreover, was intended only for himself. He could not have counted on its publication by his son Samuel.

Book cover of the version of the Arithmetica of Diophantos published by Pierre de Fermat's son Clément-Samuel in 1670 with his father's remarks


This page of the Arithmetica of 1670 contains Pierre de Fermat's marginal note