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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.
The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002, in an article titled "PRIMES is in P". [1]
Fermat's little theorem states that if p is prime and a is not divisible by p, then a p − 1 ≡ 1 ( mod p ) . {\displaystyle a^{p-1}\equiv 1{\pmod {p}}.} If one wants to test whether p is prime, then we can pick random integers a not divisible by p and see whether the congruence holds.
This must always hold if n is prime; if not, we have found more than two square roots of −1 and proved that n is composite. This is only possible if n ≡ 1 (mod 4), and we pass probable prime tests with two or more bases a such that a d ≢ ±1 (mod n), but it is an inexpensive addition to the basic Miller-Rabin test.
In Python, the NZMATH [23] library has the strong pseudoprime and Lucas tests, but does not have a combined function. The SymPy [24] library does implement this. As of 6.2.0, GNU Multiple Precision Arithmetic Library's mpz_probab_prime_p function uses a strong Lucas test and a Miller–Rabin test; previous versions did not make use of Baillie ...
Let n be a positive integer. If there exists an integer a, 1 < a < n, such that and for every prime factor q of n − 1 / ()then n is prime. If no such number a exists, then n is either 1, 2, or composite.
The Mersenne number M 3 = 2 3 −1 = 7 is prime. The Lucas–Lehmer test verifies this as follows. Initially s is set to 4 and then is updated 3−2 = 1 time: s ← ((4 × 4) − 2) mod 7 = 0. Since the final value of s is 0, the conclusion is that M 3 is prime. On the other hand, M 11 = 2047 = 23 × 89 is not prime
Otherwise, p is a strong probable prime to base a; that is, it may be prime or not. If p is composite, the probability that the test declares it a strong probable prime anyway is at most 1 ⁄ 4, in which case p is a strong pseudoprime, and a is a strong liar.