Prime number probable

In Arithmetic modular, a prime number probable (NPP) is a whole which satisfies a condition which is satisfied also by all the prime numbers. These prime numbers probable can be made up (called pseudopremiers), they are rare, dependant on the test used.

The test of Fermat for the composition, which is based on the Petit theorem of Fermat states what follows: that is to say a whole given N , let us choose a certain entirety has first with N and calculate a^{N − 1} modulo N . If the result is different from 1, N is made up. If it is equal to 1, N is first or not; N is then called a number first probable weak basic has .

The test of Fermat can be improved by the use owing to the fact that only the square roots of 1 modulo a prime number is 1 and −1. The numbers indicated as first by this reinforced test are known as numbers first probable basic forts has .

A prime number of Euler probable is an entirety which is indicated first by the somewhat reinforced theorem which affirms that: for any prime number p , and any has , has ^{( p − 1) /2} = $\ left (\ frac {has} {p} \ right)$ modulo p , where $\ left (\ frac {has} {p} \ right)$ is the Symbole of Legendre. This test is also effective and it is at least twice more extremely than the test of Fermat. A prime number probable of Euler which is made up is called a Pseudopremier d' Euler-Jacobi.

The probable primality is a base for the algorithms of test of efficiency of primality, which find an application in Cryptologie. These algorithms are generally probabilist of nature. The idea is that then there exist made up bases probably has first for any has fixed, we can reverse the roles: for any made up N fixed and a has chosen by chance, we can hope that N is not a base has pseudopremière, with a high probability.

This is unfortunately false for the probable prime numbers weak, because there exist the numbers of Carmichaël; but that is true for more refined concepts of probable primality, such as the probable prime numbers strong (we then obtain the algorithm of Miller-Rabin), or the prime numbers probable of Euler (algorithm of Solovay-Strassen).

Even when a proof of primality deterministic is necessary, a very useful stage is the test by probable primality.

A test NP is sometimes combined with a table of small pseudopremiers to quickly establish the primality of a given number smaller than a certain threshold.

See too

External bonds

http://primes.utm.edu/glossary/page.php?sort=PRP The precedes glossary - Probable premium
http://www.primenumbers.net/prptop/ The PRP Top 10000 (the largest known probable premiums)

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