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The first deterministic primality test significantly faster than the naive methods was the cyclotomy test; its runtime can be proven to be O((log n) c log log log n), where n is the number to test for primality and c is a constant independent of n. Many further improvements were made, but none could be proven to have polynomial running time.
Caldwell [10] points out that strong probable prime tests to different bases sometimes provide an additional primality test. Just as the strong test checks for the existence of more than two square roots of 1 modulo n, two such tests can sometimes check for the existence of more than two square roots of −1.
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 reason for the correctness of this claim is as follows: if the first equivalence holds for a, we can deduce that a and n are coprime.
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]
Prime95, also distributed as the command-line utility mprime for FreeBSD and Linux, is a freeware application written by George Woltman.It is the official client of the Great Internet Mersenne Prime Search (GIMPS), a volunteer computing project dedicated to searching for Mersenne primes.
A prime sieve works by creating a list of all integers up to a desired limit and progressively removing composite numbers (which it directly generates) until only primes are left. This is the most efficient way to obtain a large range of primes; however, to find individual primes, direct primality tests are more efficient [ citation needed ] .
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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