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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.
In computational number theory, the Lucas test is a primality test for a natural number n; it requires that the prime factors of n − 1 be already known. [ 1 ] [ 2 ] It is the basis of the Pratt certificate that gives a concise verification that n is prime.
We continue recursively in this manner until we reach a number known to be prime, such as 2. We end up with a tree of prime numbers, each associated with a witness a. For example, here is a complete Pratt certificate for the number 229: 229 (a = 6, 229 − 1 = 2 2 × 3 × 19), 2 (known prime), 3 (a = 2, 3 − 1 = 2), 2 (known prime),
This occurs for example when n is a probable prime to base a but not a strong probable prime to base a. [20]: 1402 If x is a nontrivial square root of 1 modulo n, since x 2 ≡ 1 (mod n), we know that n divides x 2 − 1 = (x − 1)(x + 1); since x ≢ ±1 (mod n), we know that n does not divide x − 1 nor x + 1.
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.
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A prime number q is a strong prime if q + 1 and q − 1 both have some large (around 500 digits) prime factors. For a safe prime q = 2p + 1, the number q − 1 naturally has a large prime factor, namely p, and so a safe prime q meets part of the criteria for being a strong prime.