Theorem of the two squares of Fermat

In Mathematical, the theorem of the two squares of Fermat treats modular Arithmétique.

It is sometimes named theorem of the two squares, theorem of Fermat or theorem of Fermat of Christmas. It is stated in the following way: a Prime number odd is the sum of two square of Nombre S whole, if and only, if it is adequate with a modulo four.

The first forms are old, Diophante of Alexandria in its book Arithmetica already speaks about entireties naps of two square perfect, about the year 300. The statement comes from Pierre de Fermat and the demonstration of Leonhard Euler.

Examples and theorem

One can notice that the prime numbers 5,13,17,29,37 and 41 are all naps of two squares of entireties, indeed:

$5\; =\; 1^2\; +\; 2^2,\; \backslash \; quad\; 13\; =\; 2^2\; +\; 3^2,\; \backslash \; quad\; 17\; =\; 1^2\; +\; 4^2,\; \backslash \; quad\; 29\; =\; 2^2\; +\; 5^2,\; \backslash \; quad\; 37\; =\; 1^2\; +\; 6^2,\; \backslash \; quad\; 41\; =\; 4^2\; +\; 5^2.$

On the other hand, 3,7,11,19,23 and 31 do not have this property. The theorem of the two squares of Fermat clarifies this situation:

Either p a prime number odd, p is nap of two squares of entireties if and only if p is adequate with 1 modulos 4.

$\backslash \; exists\; (X,\; there)\; \backslash \; in\; \backslash \; N^2\; \backslash \; quad/\backslash \; quad\; p\; =\; x^2\; +\; y^2\; \backslash \; quad\; \backslash \; yew\; \backslash \; quad\; p\; \backslash \; equiv\; 1\; \backslash \; pmod\; \{4\}\; \backslash ;\; or\; \backslash ;\; p=\; 2\; \backslash ;$

History

This theorem is conjectured by Pierre de Fermat (1601 - 1665) in a written letter with Marin Mersenne dated December 25th 1640. For this reason sometimes, one calls it “  theorem of Christmas de Fermat  ”. As often at Fermat, no proof is given.

The first proof is the work of Leonhard Euler (1707 - 1783) . She is announced in a letter addressed to Christian Goldbach in 1749. This proof is obtained by a technique called infinite descent. Joseph-Louis Lagrange (1736 - 1813) provides a new proof in 1775 using its work on the quadratic forms.

The modern evidence is based generally on the ring whole of Gauss. They are indeed simpler and more natural. The original proof using this unit is the work of Carl Friedrich Gauss (1777 - 1855) published in 1801. Richard Dedekind (1831 - 1916) proposes two elegant evidence using the entireties of Gauss.

Motivation

The equations diophantiennes, in general, fascinated the mathematicians. Their charm particular , wrote Gauss, comes from the simplicity of the statements united to the difficulty of the evidence . This general remark applies to this theorem.

The Conjecture S of Fermat have sometimes an additional property which, as from 19th century made them so famous. Their resolutions ask for the development of fertile tools in mathematics. A considerable part of the Théorie of the rings comes from the desire to better include/understand the integers.

With the difference of the Great theorem of Fermat, the theorem of the two squares is not at the origin of a development of particular tool. On the other hand, it represents a splendid example of uses of often powerful methods, and which, each time simplifies the demonstration.

This theorem comprises some applications, for example the research of the prime numbers of Gauss. The principal motivation, that which led great names of mathematics to publish evidence, remains primarily the exemplary aspect their demonstrations.

Demonstrations

Gauss and its entireties

See also: Whole of Gauss

Gauss proposes a revolutionary demonstration in 1801 in its treaty Disquisitiones Arithmeticae . It uses for that the whole which bear its name now.

An entirety of Gauss Z is a Complex number having its whole parts real and imaginary. If Z = has + I. B , one defines his standard NR ( Z ) as equal to has ² + B ². It corresponds squared of the distance from the point origin. Graphically the entireties of Gauss are the points of the squaring of the figure of right-hand side. The theorem amounts finding the intersections of the circles of center the origin and ray the square Racine of a prime number. The figure illustrates the fact that 2,5 and 13 have solutions, whereas 3,7 and 11 in do not have. For example, the point on the right of the figure illustrates the solution 3² + 2² = 13.

Across, the graphic aspect of the method, the true interest lies in the fact that the whole of the entireties of Gauss has an arithmetic with its prime numbers called irreducible numbers and its Lemme of Euclide.

This approach, makes it possible to show in the same treaty another famous conjecture, the quadratic Loi of reciprocity that Gauss regarded as the theorem of gold . It is at the origin of vast branches of mathematics, the modular Arithmétique and the Algebraic theory of the numbers.

Dedekind and modern algebra

Dedekind proposes at least two demonstrations of this theorem illustrating the use of the arithmetic properties of the entireties of Gauss. That presented here is the second, it is published in 1894 in Supplement XI of the Leçons in theory of the numbers of Dirichlet.

Its approach is elegant and expeditious. Dedekind is the last pupil of Gauss. It is based on the techniques of the arithmetic Recherches the book of its Master.

Initially uses the arithmetic modular one: a cyclotomic Polynôme on the cyclic body of cardinal p makes it possible to show the existence of an entirety α such as α² + 1 is a multiple of the prime number. The cyclic group of order four, corresponding to the group of the entireties of Gauss invertible and illustrated in green on the figure, appears immediately.

Then, it solves the arithmetic question with two remarks about the entireties of Gauss.

The demonstration illustrates perfectly the power of the theoretical tools developed during its century, especially if one compares it with that of Euler, the known first.

Lagrange and quadratic forms

Euler and infinite descent

See also: Theorem of infinite descent

The first known demonstration, that of Euler, are based on a direct approach, not using any the tools quoted above. Euler uses only the Identité of Brahmagupta, the Petit theorem of Fermat and the elementary rules of calculus. This proof illustrates the quantity of easy ways necessary at the time to come to end from the proof. It is inevitably long and comprises heavy calculations.

Euler uses at the fourth stage of its demonstration a reasoning of following nature, if the theorem is false, then it exists an entirety has different from zero which checks a property such as has is written in the form X . there each one different from one with there having the same property. has would then have an infinity of dividers different from one, which is impossible.

Such a step is called an infinite descent. She plays a big role into arithmetic. Fermat uses it to show the Grand theorem of Fermat for N equal to four. Euler uses it to propose a demonstration of the case N equal to three, the demonstration proves to be false, the approach is nevertheless the good one. It allows even the resolution of the case where N is equal to five.

A modern demonstration

This theorem has many different evidence. An analytical approach allows, for example, a demonstration using only little of tools and relatively simple nevertheless.

Related results

Other problems arising from Fermat

Fourteen years later, in a letter with Blaise Pascal, Fermat conjectures two similar results if p is a prime number odd:

• $p\; =\; x^2\; +\; 2y^2\; \backslash \; Leftrightarrow\; p\; \backslash \; equiv\; 1\; \backslash \; mbox\; \{or\}\; p\; \backslash \; equiv\; 3\; \backslash \; pmod\; \{8\}$

• $Si\; \backslash ;\; p\; \backslash \; neq\; 3\; \backslash \; quad\; then\; \backslash \; quad\; p=\; x^2\; +\; 3y^2\; \backslash \; Leftrightarrow\; p\; \backslash \; equiv\; 1\; \backslash \; pmod\; \{3\}$

These two results for the first time are shown by Lagrange.

Generalization with all the entireties

Once known the prime numbers nap of two squares, it becomes possible to generalize the question with all the entireties:

* an entirety N is nap of two squares of entireties if, and only if, in its decomposition in factors first, the prime numbers adequate with three modulos four appear in an even power.

This result can be stated in the following way:

An entirety is nap of two squares of entireties if and only if the p-adic valuations of the factors first p of N adequate with three modulos four are even.

Arithmetic functions

The use of the arithmetic functions makes it possible to obtain more results. Thus, if one poses $r\_2\; (N)$ as being the number of ways of writing the whole in the form of a sum of 2 squares, one can prove the following result:

$r\_2\; (N)\; =\; 4\; \backslash \; left\; (d\_1\; (N)\; -\; d\_3\; (N)\; \backslash \; right)$.

• $d\_1\; (N)$ indicates the number of dividing $d$ of $n$ checking $d\; \backslash \; equiv\; 1$.

• $d\_3\; (N)$ indicates the number of dividing $d$ of $n$ checking $d\; \backslash \; equiv\; 3$.

See too

External bonds

• Somme of squares per G. Yoda
• Somme of two squares by Bibmath

References

• I.R.E.M. Lille numbers - old and current Problems Ellispe 2000 ISBN 2_7298_0122_7

Random links:

Painting of the Rebirth | 35007 | Park TGV-Futuroscope | Action of responsibility | Accounts of the Stål ensign | Liste_de_films_de_zombi

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org