Diviseur

Résumé

Le mot “diviseur” a deux significations en mathématiques. Une division est effectuée à partir d’un “dividende” et d’un “diviseur”, et une fois l’opération terminée, le produit du “quotient” par le diviseur augmenté du “reste” est égal au dividende. En arithmétique, un “diviseur” d'un entier n est un entier dont n est un multiple. Plus formellement, si d et n sont deux entiers, d est un diviseur de n seulement s'il existe un entier k tel que . Ainsi est un diviseur de car . La notion de diviseur est liée à celle de multiple, car si d divise n alors n est un multiple de d, et à la notion de divisibilité. Le nom vient de l'opération arithmétique de division : si a, b sont des entiers avec b non nul, et si c = a/b est un entier, alors a est le dividende, b le diviseur et c le quotient. Ensemble des diviseurs d'un entier Si l'entier n est nul, tout entier divise n. Si l'entier n est non nul, il possède des diviseurs positifs et négatifs, mais pas de diviseur nul. Si d est

Source officielle

https://fr.wikipedia.org/wiki/Diviseur

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Primality Testing

Kamal Tahiri Jouti

Since the discovery of the utility of the numbers, the human being tried to differentiate them. We decide between them according to whether they are even or odd. Or, according to the fact that they are prime or composite. A natural number n >1 is called a prime number if it has no positive divisors other than 1 and n. Therefore, other numbers that are not prime have other divisors. That is why we call them composite numbers because we can write them : n = p*q with {p,q} ≠ {1,n}. The problem has always been to decide whether a number is prime or not. To answer this problem, many algorithms have been created like the Trial Division. It uses the property which says that the biggest divisor of n is smaller or equal to the square root of n. But for numbers that exceed 30 digits, it will take more than 10^13 years to know the answer. So, the interest would be to create an algorithm using mathematical bases which would answer to this question as fast as possible. This is what we will see in this project. The study of prime numbers became really important to code texts. Cryptography is one of the most important application of prime numbers theory. At the beginning, it was only used to code texts during the wars and more recently it was used for other applications, like the security of an account. Fist of all, I will focus on randomized algorithms for primality testing. Then, I will focus on a deterministic algorithm that I have implemented.

2006

Chargement

We take an approach toward Counting the number of integers n for which the curve (n),: y(2) = x(3) - n(2)x has 2-Selmer groups of a given size. This question was also discussed in a pair of papers by Roger Heath-Brown. In contrast to earlier work, our analysis focuses oil restricting the number of prime factors of n. Additionally, we discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is the "independence" of the Legendre symbol evaluated at the prime divisors of an integer with exactly k prime factors. (C) 2009 Elsevier Inc. All rights reserved.

2009

Chargement

Statistics Of Prime Divisors In Function Fields

Robert Christopher Rhoades

We show that the prime divisors of a random polynomial in F-q[t] are typically "Poisson distributed". This result is analogous to the result in Z of Granville [1]. Along the way, we use a sieve developed by Granville and Soundararajan [2] to give a simple proof of the Erdos-Kac theorem in the function field setting. This approach gives stronger results about the moments of the sequence {omega(f)}(f is an element of Fq)[t] than was previously known, where omega(f) is the number of prime divisors of f.

2009

Chargement