Small theorem of Fermat

In Mathematical, the small theorem of Fermat is a result of the modular Arithmétique.

It is stated as follows. If has is an unspecified entirety and p a Prime number, then has ^p - has is a multiple of p .

It owes its name with Pierre de Fermat (1601 - 1665) which states it the first time the October 18th 1640.

It has many applications, at the same time in modular Arithmétique and Cryptographie.

History

The China seems to be the first culture to be itself interested in the modular Arithmétique. There exists an assumption, refuted by Joseph Needham, according to which numbers of the form 2 ^p - 2 has been studied by this civilization for 2.500 years.

The first known appearance of this theorem comes from a letter of Fermat with Bernard Frénicle de Bessy (1605 - 1675) . One can read this there: " Any prime number the measurement infallibly one of powers -1 of some progression that it is, and the exhibitor of the aforesaid the power is submultiple of the prime number given -1… ". This formulation is the exact equivalent one of the modern formulation given in introduction. Fermat had probably shown this result, it specifies indeed in its letter: " And this proposal is generally true in all progressions and all prime numbers; of what I you envoierois the demonstration, if I appréhendois to be too much long".

At that time, it is of use not to publish the evidence of the theorems. Thus Gottfried Wilhelm von Leibniz (1646 - 1716) writes a demonstration towards 1683 but does not publish it. The proof becomes public in 1736 following work of the mathematician Leonhard Euler (1707 - 1783) . Carl Friedrich Gauss (1777 - 1855) writes a new faster proof in 1801.

The term commonly used until the 20th century is theorem of selected Fermat for example per Gauss in its book Disquisitiones arithmeticae. The theorem changes name into 1913 to take its current form.

Examples

Here some examples of the theorem:

• 5³ − 5 = 120 is divisible by 3.
• 7² − 7 = 42 is divisible by 2.
• 2⁵ − 2 = 30 is divisible by 5.
• (− 3) ⁷ + 3 = − 2.184 is divisible by 7.
• 2⁹⁷ − 2 = 158.456.325 028.528.675 187.087.900 670 is divisible by 97.

Applications

The applications are numerous, particularly in Cryptographie. One finds nevertheless examples traditional of applications of the theorems in pure Mathématiques.

Theoretical applications

The small theorem of Fermat is historically used to analyze the Décomposition in product of factors first of certain entireties. Thus Fermat written with Marine Mersenne (1588 - 1648) : " You ask to me whether number 100.895.598.169 is first or not, and a method to discover, in the one day space, if it is first or made up. With this question, I answer that this number is composed and is made product from these two: 898.423 and 112.303, which is premiers". By using a similar method, Euler cancels the single false conjecture of Fermat, namely that the numbers of Fermat are not very first.

This theorem is also used to show results of Algebraic theory of the numbers, like the Théorème of Herbrand-Ribet. It exceeds the strict framework of the Arithmétique, with a use for the study of the fixed points of the endomorphism of Frobenius for example.

Asymmetrical cryptography

See also: asymmetrical Cryptography

Cryptography with public key corresponds to a code attempting to ensure the Confidentialité key messages using two of coding. One, allowing to quantify the message, is public. The other having for objective decoding is kept secret.

An important family of codes asymmetrical uses technology called Rivest Shamir Adleman. The secret key is the data decomposition of a great integer, often of several hundreds of decimals. It contains two factors first. The main part of the techniques industrialists of the beginning of the 21e century is based on the small theorem of Fermat to generate great prime numbers or to test the primality of a number.

Test of primality

See also: Test of primality

The small theorem of Fermat gives a requirement so that a number p is first. It is necessary indeed that, for all has smaller than p has ^{p - 1} is adequate with a modulo p . This principle is the base of the Test of primality of Fermat. There exist many alternatives algorithmic of this test. Most known are the Test of primality of Solovay-Strassen and especially the Test of primality of Miller-Rabin.

Pseudo-first number

See also: Number pseudopremier

The preceding tests use a nonsufficient requirement but. Thus, there exist entireties p not first such as for all has ranging between a and p - 1 , has ^{p - 1} is always adequate with a modulo p . The number 1729 is an example. Such numbers are called Nombre of Carmichaël.

The tests indicated at the preceding paragraph are all statistical, with the direction or there exists always a probability, sometimes very weak for the number having passed the test is not nevertheless first. These numbers are called pseudopremiers and are attached to particular tests.

Generalizations

A light generalization of the theorem, which rises immediately from this one, states   in the following way: if p is a prime number and if m and N is strictly positive entireties such as m ≡ N (MOD p -1), then for all entireties has , has ^m ≡ has ^N (MOD p ). In this form, the theorem is used to justify the algorithm of figuring RSA with public key.

The small theorem of Fermat is generalized by the Théorème of Euler  : for entire naturalness not no one N and entire has first with N , one has

$a^\; \{\backslash \; varphi\; (N)\}\; \backslash \; equiv\; 1\; \backslash \; pmod\; \{N\}$

where φ ( N ) cash indicates the function φ of Euler the entireties between 1 and N which is first with N . If N is a prime number, then φ ( p ) = p - 1, one finds the small theorem of Fermat.

The demonstration is based on the fact that the Groupe of the units of the Anneau Z/nZ is of order φ ( N ).

Demonstrations

Arithmetic modular

See also: Arithmetic modular

The knowledge of the structure and particularly of the group of the units of the ring '' Z ''/'' pZ '', allows a simple demonstration of the theorem. If p is a prime number, the group of the units Z / pZ ^* is a cyclic Groupe of order p - 1, therefore isomorphous with Z /( p - 1) Z .

A first approach consists in considering φ this isomorphism. The image of φ ( has ^p-1) is equal to ( p - 1) φ ( has ), corresponding to the neutral element of the group. One deduced that has ^p-1 is the neutral element of Z / pZ ^*, i.e. unit classifies it, which finishes the demonstration.

A second approach is the application of the theorem of Lagrange, the order of any element of a finished group is a divider about the group. Consequently, if θ is the order of has , then there exists an entirety μ such as θ.μ = p - 1. The entirety has ^θ is an element of the class of the unit per definition about an element (cf the paragraph Définitions of the cyclic article Groupe) and thus has ^{p - 1}= has ^θ.μ is also element of the class of the unit.

These approaches correspond at the same time to the work of Gauss and with the modern demonstrations, they are indeed most concise.

Demonstration of Euler and Leibniz

There exists another demonstration, using the Formule of the binomial theorem. This demonstration corresponds to that of Euler and Leibniz.

It uses a Raisonnement by recurrence on the value has . On account of the simplicity, the notations used here are that of Gauss, using congruences. If these notations do not correspond to those of the time, the reasoning is nevertheless identical.

A demonstration limited to the knowledge of Fermat

It is possible to imagine a demonstration calling only upon knowledge of Fermat. It is longer consequence and asks a more astute step to succeed. It uses primarily the Lemme of Euclide, the Euclidean Division and the Identité of Bézout.

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