Demonstrations of the last theorem of Fermat
In Mathematical, more precisely in Arithmetic modular, the Grand theorem of Fermat treats roots of the following equation diophantienne:
In 1637 Pierre de Fermat (1601 - 1665) proposes this equation and indicates in the margin of its specimen of the book Arithmetica of Diophante which it found a marvellous demonstration .
It is not very probable that a demonstration accessible to Fermat exists, one needs many attempts as nearly 350 years of efforts so that a proof is given by it in 1994 by Andrew Wiles.
General information
Elementary remarks
The equation is homogeneous, i.e. for a value N given, if the triplet ( X , there , Z ) is solution, then ( has . X , has . there , has . Z ) is also solution. Consequently, the only required roots are the whole triplets of first between them as a whole.If one of the three members of the triplet ( X , there , Z ) is equal to zero, then the equation becomes obvious, such solutions are known as commonplace. The objective is thus the search for triplet solution such as the product X . there . Z is different from zero.
If the equation does not admit a solution for a value p of the parameter, then there does not exist solution for any value multiple N of p . Indeed, if one notes N = p . Q then the equation is written:
Results without call to the theory of the numbers
Some results are shown without complex structure. The case where N equal to two, is treated after, is simple and dates from antiquity. That where N is equal to four shows in the very small little least elementary way. The cases remaining are those where N is first different from two. There exists a demonstration which does not use the whole of Eisenstein for the case where N is equal to three , it is nevertheless sufficiently astute and difficult so that the mathematician Leonhard Euler proposes only one inaccurate demonstration.The other cases are technical, the use of whole algebraic is essential. The first term is well a remarkable identity: X n + n is there indeed a multiple of X + there if N is not a power of two , however this remark is largely insufficient to conclude would be this only in one case.
Case where N is equal to two
See also: Triplet pythagorician
The case where N is equal to two has a geometrical interpretation. It corresponds to the whole lengths of different with dimensions from a right-angled triangle.
This case is known since high antiquity. Thus, the Sumérien S knew some example of solutions. The total solution appears for the first time in book X of the Eléments of Euclide towards 300 av J.C.
This case is the single exception of the theorem (if one omits the case where N is equal to one). Indeed, there exist noncommonplace solutions if N is equal to two. Three four and five is a triplet of solutions, such a triplet is called triplet pythagorician . Consequently, it becomes important to consider the case N equal to four, to show that there does not exist other power of two admitting noncommonplace solutions.
A demonstration is given in the article Triplet pythagorician.
Case where N is equal to four
This case is probably the single treaty by Fermat. If no documentary evidence is found in its correspondence, on the other hand it shows that there does not exist any triplet pythagorician such as X . /2 is a square there of entirety, which is expressed, in the vocabulary of the author by the surface of a Right-angled triangle cannot be that of a Carré . From this result, the demonstration is easy.The method used is that of the infinite Descente. The method consists in finding another triplet of solutions such as the third entirety is positive and strictly smaller than that of the initial solution. It is thus possible indefinitely to descend as a whole from the positive entireties, which is contradictory with the properties of NR .
The final proof comes from Leonhard Euler (1707 - 1783) , it is also founded on the method of the infinite descent. There is the different one, for example using the concept of whole of Gauss
Case where N is equal to three
See also: Whole of Eisenstein
The case is more complex, Euler writes with Goldbach in 1753, indicating to him that it solved this case. It publishes its proof, which appears false. For its demonstration, he studies numbers whose cube is form p 2 + 3. Q 2, for that it uses an original method for the time, it considers the unit Z , and treats this whole like a factorial Anneau, i.e. it supposes the unicity of a writing of an element in irreducible elements. This result is not exact, for example 4 is at the same time equal to 2 X 2 and also with (1 + √3.i) (1 - √3.i).
Thirty years later, Carl Friedrich Gauss (1777 1855) publishes a treaty where, for the first time, a ring of whole algebraic is studied rigorously, the ring of the entireties which bears its name now. The logic of the modular Arithmétique becomes applicable and a demonstration similar to that of Euler is now rigorous.
The algebraic ring of entirety making it possible to simply analyze the case where the parameter is equal to three is studied precisely by Ferdinand Eisenstein (1823 1852) . This unit is equal to Z where J indicates the cubic root of the unit having a pure imaginary part strictly positive. This ring is Euclidean thus factorial, consequently, the decomposition in irreducible elements so called prime numbers of Eisenstein is quite single.
The use of quite selected ring of entireties is one of the major techniques of the 19th century. For the resolution of the theorem with certain parameters. On the other hand, rare are the rings of Euclidean entireties. Other techniques must then be assistant to arrive at a certain general information.
In this ring of entireties called whole of Eisenstein, an infinite descent is relatively simple to find, it is the method used in the proof suggested here.
German theorem of Sophie
The step making it possible to solve the case where N is equal to three does not generalize. Indeed, the ring of the algebraic whole associated with the roots of the units is not factorial any more. The arithmetic reasoning of the preceding case is not thus operational any more.
During the first decade of the 19th century, Sophie Germain (1776 - 1831) brings a new idea called
- Théorème of Sophie Germain : If N > 2 and 2 N + 1 is prime numbers and if X . there . Z is not multiple of N , then the triplet ( X , there , Z ) is not solution of the equation of Fermat.
The demonstration of the case where the parameter is equal to three uses a step of this nature.
The study of the demonstration of the theorem is then divided into two cases:
- * There exists a value of the multiple triplet of N .
- * None members is multiple of N .
Sophie Germain solves the first case for all the values of the parameter lower than hundred. Adrien-Marie Legendre (1752 - 1833) pushes the demonstration with all the values smaller than a hundred and four twenty seventeen.
Case where N is equal to five
See also: Whole of Dirichlet
The theorem of Fermat is then famous. All the efforts concentrate on the case where the parameter is equal to five. Sophie Germain solved the case where none the unknown factors is multiple of five. However, in spite of the implication of many members of the mathematical community, more than fifteen years run out without notable progress. In 1825 Dirichlet (1805 - 1859) becomes immediately famous, for a significant contribution. In general, a triplet solution, if N is equal to five, contains a multiple of five and one multiple of two. Dirichlet solved the case where the two multiples are associated with the same unknown factor.
The demonstration is subjected to the Academy of Science and Legendre is named referred. It uses the techniques of Dirichlet, and solves the other case in a few months, i.e. that where the multiple of two and that of five are associated with different unknown factors.
The two demonstrations use techniques similar to that of the case where the parameter is equal to three. They are based on arithmetic modular ring of the relative entireties and a quite selected ring of entireties. However, this time Ci and with the difference in the case where N is equal to three, the associated cyclotomic extension, i.e. corresponding to the Corps of decomposition of the cyclotomic Polynôme is neither Euclidean nor factorial. It becomes necessary to consider the ring of the entireties of the body Q and not Q . The structure, particularly the Groupe of the units becomes more complex. Its comprehension returns to the analysis of another equation diophantienne known as of Pell, studied by Euler. Work of Lagrange on the continuous fractions provides the necessary tools to the elucidation of this structure. This ring takes the name of ring of the Entier of Dirichlet, it makes it possible to establish the key lemma of the demonstration.
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With the difference in work of Gauss and Eisenstein on the case where N is equal to three, no major theoretical opening is carried out for the resolution of this case. The associated ring is always Euclidean and thus factorial, the arithmetic ones used are of comparable nature that the preceding ones.
See too
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