Primality test

Summary

A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input). Some primality tests prove that a number is prime, while others like Miller–Rabin prove that a number is composite. Therefore, the latter might more accurately be called compositeness tests instead of primality tests. Simple methods The simplest primality test is trial division: given an input number, n, check whether it is divisible by any prime number between 2 and \sqrt n (i.e., whether the division leaves no remainder). If so, then n is composite. Otherwise, it is prime. In fac

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https://en.wikipedia.org/wiki/Primality_test

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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

Randomized and Deterministic Primality Testing

Monica Perrenoud

2009

Experiments with the Markoff Surface

Matthew De Courcy-Ireland, Sandy Lee

We confirm, for the primes up to 3000, the conjecture of Bourgain-Gamburd-Sarnak and Baragar on strong approximation for the Markoff surface modulo primes. For primes congruent to 3 modulo 4, we find data suggesting that some natural graphs constructed from this equation are asymptotically Ramanujan. For primes congruent to 1 modulo 4, the data suggest a weaker spectral gap. In both cases, there is close agreement with the Kesten-McKay law for the density of states for random 3-regular graphs. We also study the connectedness of other level sets . In the degenerate case of the Cayley cubic, we give a complete description of the orbits.

TAYLOR & FRANCIS INC2020