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### Abstract

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.