Last theorem of Fermat

In Mathématiques, the last theorem of Fermat , or theorem of Fermat-Wiles , states that there is no integers nonnull $x$ , $y$ and $z$ such as:

x^n+y^n=z^n \,

as soon as

n

is an entirety strictly higher than 2 .

For the values of $n$ lower or equal to 2, there exists an infinity of solutions. The case $n=1$ is obvious. The case $n=2$ admits in particular the traditional solution 3² + 4² = 5². In a general way, all the solutions for $n=2$ are given by:

$x=2kml \,$

y=k (m^2-l^2) \,

z=k (m^2+l^2) \,

where the numbers K, L and m satisfy the conditions: K whole, m>l, m and L of different parities. One calls sometimes these entireties the triplets pythagoricians.

However, as soon as $n$ is higher than two, it is not possible any more. The Théorème owes its name with Pierre de Fermat which wrote in margin of a translation of the Arithmetica of Diophante, beside the statement of this problem:

I found a marvellous demonstration of this proposal, but the margin is too narrow to contain it.

In Latin: Cubum autem in duets cubos, aut quadratoquadratum in duets quadratoquadratos, and generaliter nullam in infinitum ultra quadratum potestatem in duets ejusdem nominis fas is dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas not caperet.

This sibylline note let think that an elementary demonstration was possible; what thus highly émoustillé the curiosity of people. After having been the object of feverish research during more than 300 years, the theorem was finally shown in 1994 by the mathematician Andrew Wiles. The majority of the mathematicians think today that Fermat had been mistaken while thinking correctly of having shown its Conjecture: the known proof (refined since) called upon very powerful tools of Theory of the numbers.

More precisely, Wiles proved a particular case of the conjecture of Shimura-Taniyama-Weil, which one for some time knew already that it implied the theorem. The proof calls upon the modular forms, with representations galoisiennes… The proof passed by the establishment partial of the conjecture of Shimura-Taniyama whose implications for the theorem of Fermat followed ideas of Yves Hellegouarch, Gerhard Frey, Jean-Pierre Serre and Ken Ribet.

This theorem does not have any application in oneself: it is by the ideas that it was necessary to implement to show it, by the tools which were installed with this intention, that it takes such a value. The article Demonstrations of the last theorem of Fermat watch some examples of tools discovered and used for the resolution of this problem.

One can also include/understand this theorem graphically by considering the curve of equation xⁿ + yⁿ = 1: if n > 2, this curve does not pass by any point to nonnull rational coordinates.

Remarks

The use wanting that one gives to a Théorème the name of that which brought the demonstration from there, the name of “theorem of Fermat” is not justified strictly speaking. It would be necessary to speak either about a “conjecture of Fermat”, or of the “theorem of Wiles”.

As opposed to what one has been able sometimes to see in newspapers or with Television (because of the type of formula which appears there), this theorem does not have really a relation with the Théorème of Pythagore. In fact, the object of the theorem of Pythagore is to give a geometrical characterization of the triangles pythagoricians, i.e. of which the lengths on the sides form a triplet pythagorician, these triplets being themselves solutions of the equation of Fermat in the case N = 2. The analogy with the theorem of Fermat is thus the question of the existence of triplets pythagoricians, and the question of their geometrical interpretation is clearly another question. Let us notice nevertheless that Fermat obviously took as a starting point the concept of triplet pythagorician: its conjecture is indeed noted in margin of a talk of Diophante about the triplets pythagoricians.

References

Wiles, Andrew (1995). Modular elliptic curves and Fermat' S last theorem, Annals off Mathematics ( 141 ) (3), 443-551
Simon Singh, the Last Theorem of Fermat ,

See too

Demonstrations of the last theorem of Fermat

Be-X-old: ВялікаятэарэмаФэрма Simple: Fermat' S last theorem

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