Fermat's Last Theorem

Pierre de Fermat.

In number theory, Fermat's Last Theorem, or Fermat-Wiles theorem, is one of the most famous theorems in the history of mathematics. Using modern notation, it can be stated as follows:

This is true except for the trivial solutions (0,1,1), (1,0,1) and (0,0,0). It is important to emphasize that they must be positive, since if any of them could be negative, it is not difficult to find non-trivial solutions for a case in which n is greater than 2. For example, if n were any odd number, the triples of the form (a, -a, 0) with a a positive integer, are solutions.

This theorem was conjectured by Pierre de Fermat in 1637, but it was not proved until 1995 by Andrew Wiles aided by the mathematician Richard Taylor. The search for a proof stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century.

Historical introduction

Arithmetic Diofanto. The 1670 edition includes Fermat's comments; the one under problem VIII is known as his "last theorem".

Pierre de Fermat owned a bilingual (Greek and Latin) edition of Diophantus's Arithmetica, translated by Claude Gaspar Bachet. Fermat wrote a commentary, in fact a riddle, in the margin of each problem, and one by one they have been solved by personalities such as Leibniz, Newton, etc. He only left unsolved the puzzle that he proposed under problem VIII, which deals with writing a square number as the sum of two squares (ie, finding Pythagorean triples). There, Fermat wrote:

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 marginalis exiguitas non caperet.

It is impossible to break a cube into two cubes, a two-quartered, and in general, any power, apart from the square, into two powers of the same exponent. I found a really admirable demonstration, but the book margin is very small to put it.

Pierre de Fermat

History of the proof of the theorem

Pierre de Fermat

The first mathematician who managed to advance on this theorem was Fermat himself, who proved the case n=4 using the technique of infinite descent, a variant of the principle of induction.

Leonhard Euler

Leonhard Euler proved the case n = 3. On August 4, 1753 Euler wrote to Goldbach claiming to have a proof for the case n = 3. In Algebra (1770) found a fallacy in Euler's proof. Correcting it directly was too difficult, but Euler's earlier contributions allowed a correct solution to be found by simpler means. This is why Euler was considered to have proved that case. From the analysis of Euler's failed proof emerged evidence that certain sets of complex numbers did not behave in the same way as integers.

Sophie Germain

The next major step was made by the mathematician Sophie Germain. A special case says that if p and 2p + 1 are both prime, then the expression of Fermat's conjecture for the power p implies that one of the x, y, or z is divisible by p. Consequently the conjecture is divided into two cases:

• Case 1: None of the x, and, z It's divisible by p;
• Case 2: one and only one x, and, z It's divisible by p.

Sophie Germain proved case 1 for all p less than 100 and Adrien-Marie Legendre extended her methods to all numbers less than 197. Here case 2 was found not even to be proven. for p = 5, so it was clear that case 2 was the one to focus on. This case was also divided among several possible cases.

Ernst Kummer and others

Chronology

Year	Development
1665	Muere Fermat without registering his demonstration.
1753	Leonhard Euler demonstrated the case n=3{displaystyle n=3}.
1825	Adrien-Marie Legendre showed the case for n=5{displaystyle n=5}.
1839	Lamé demonstrated the case n=7.
1843	Ernst Kummer claims to have shown theorem but Dirichlet finds a mistake.
1995	Andrew Wiles publishes the theorem demonstration.

It was not until 1825 that Peter Gustav Lejeune Dirichlet and Legendre generalized Euler's proof to n=5. Lamé proved the case n=7 in 1839.

Between 1844 and 1846 Ernst Kummer showed that non-unique factorization could be overcome by introducing ideal complex numbers. A year later Kummer affirms that the number 37 is not a regular prime (See: Bernoulli numbers). Then it is found that neither are 59 and 67. Kummer, Mirimanoff, Wieferich, Furtwänger, Vandiver, and others extend the investigation to larger numbers. In 1915 Jensen proved that there are infinitely many irregular primes. Research stagnates along this path of divisibility, despite the fact that tests are achieved for n less than or equal to 4 000 000.

Andrew Wiles

In 1995 the mathematician Andrew Wiles, in a 98-page article published in the Annals of mathematics, proved the semi-stable case of the Taniyama-Shimura theorem, previously a conjecture, which links the forms modular and elliptical curves. From this work, combined with Frey's ideas and Ribet's theorem, the proof of Fermat's last theorem follows. Although an earlier (unpublished) version of Wiles's work contained an error, this could be corrected in the version published, consisting of two articles, the second in collaboration with the mathematician Richard Taylor. In these works, for the first time, modularity results are established from residual modularity, for which the results of the type proven by Wiles and Taylor are called «modular survey theorems». At present, results of this type, much more general and powerful, have been proved by several mathematicians: in addition to generalizations proved by Wiles in collaboration with C. Skinner and by Taylor in collaboration with M. Harris, the most general at present. are due to Mark Kisin. In Wiles's 1995 work, a new path was opened, practically a new area: that of modularity. With these techniques, of which this work was a pioneer, other important conjectures have been solved more recently, such as the Serre conjecture and the Sato-Tate conjecture. Curiously, the resolution of the first cases of Serre's conjecture (works by Khare, Wintenberger and Dieulefait), as Serre himself observed when formulating the conjecture, allows a new proof of Fermat's last theorem.

Wiles's work therefore has an importance that far transcends its application to Fermat's Last Theorem: it is considered central to modern arithmetic geometry and is expected to continue to play a vital role in the proof of modularity results to be framed in the Langlands program.