Fermat’s theorem

Fermat’s Last Theorem . It was formulated by Pierre de Fermat in 1637 and its proof was still achieved, 358 years after its proposal. The multi-century and multi-national effort to find proof of the conjecture still drove the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century.

Summary

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  • 1 Story
  • 2 Timeline of the demonstration
  • 3 Sources
  • 4 References and notes
  • 5 External Links

History

Diofantica Problem VIII

In the book Arithmetica de Diofanto, Fermat wrote comments on the margins of each problem, and one by one they have been solved by personalities such as Leibniz, Newton, etc. Only the riddle he proposed under problem VIII, which tries to find a square number as the sum of two squares, known as Pythagorean triples, remained unsolved.

Fermat’s note said: It is impossible to decompose a cube into two cubes, a quadratic cube into two cubic cubes, and in general, any power, apart from the square, into two powers of the same exponent. I have found a really admirable demonstration, but the margin of the book is too small to put it.

Demonstration timeline

Several were those who tried to demonstrate the VIII problem of the book Arithmetica de Diofanto, among them;

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

Leonhard Euler proved case n = 3. On August 4, 1735 Euler wrote to Goldbach claiming to have a proof for case n = 3. In Algebra (1770) a fallacy was found in Euler’s proof. Correcting it directly was too difficult, but Euler’s earlier contributions made it possible to find a correct solution by simpler means. That is why Euler was considered to have demonstrated that case. Evidence emerged from Euler’s failed proof analysis that certain sets of complex numbers did not behave the same as integers.

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

  • Case 1: none of the x, y, z is divisible by p; • Case 2: one and only one of x, y, z is divisible by p.

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

Ernst Kummer and others It was not until 1825 that Peter Gustav Lejeune Dirichlet and Legendre generalized Euler’s proof to n = 5. Lamé demonstrated the case n = 7 in 1839 .

Between 1844 and 1846 Ernst Kummer demonstrated that non-unique factorization could be saved by introducing ideal complex numbers . A year later Kummer claims that number 37 is not a regular cousin .

Then it is found that 59 and 67 are not either. Kummer, Mirimanoff, Wieferich, Furtwänger, Vandiver, and others extend the investigation to larger numbers. In 1915 Jensen demonstrates that infinite irregular primes exist. Research stagnates along this path of divisibility, even though checks are made for n less than or equal to 4,000,000.

Andrew Wiles In 1995 , mathematician Andrew Wiles , in a 98-page article published in Annals of mathematics, demonstrated the semi-stable case of the Taniyama-Shimura Theorem , formerly a conjecture, that links modular shapes and elliptic 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’ work contained an error, it could be corrected in the published version, which consists of two articles, the second in collaboration with 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”. Today, much more general and powerful results of this kind have been tested by various mathematicians: in addition to generalizations tested by Wiles in collaboration with C. Skinner and Taylor in collaboration with M. Harris, the most general ones today are they owe Mark Kisin .

In Wiles’ 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 the Serre Conjecture, 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 goes far beyond its application to Fermat’s Last Theorem: they are considered central to modern Arithmetic Geometry and are expected to continue to play a vital role in demonstrating modularity results that are framed in the Langlands Program .

 

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