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Prime95 tests numbers for primality using the Fermat primality test (referred to internally as PRP, or "probable prime"). For much of its history, it used the Lucas–Lehmer primality test, but the availability of Lucas–Lehmer assignments was deprecated in April 2021 [7] to increase search throughput. Specifically, to guard against faulty ...
Otherwise, n may or may not be prime. The Solovay–Strassen test is an Euler probable prime test (see PSW [3] page 1003). For each individual value of a, the Solovay–Strassen test is weaker than the Miller–Rabin test. For example, if n = 1905 and a = 2, then the Miller-Rabin test shows that n is composite, but the Solovay–Strassen test ...
All Mersenne primes are of the form M p = 2 p − 1, where p is a prime number itself. The smallest Mersenne prime in this table is 2 1398269 − 1. The first column is the rank of the Mersenne prime in the (ordered) sequence of all Mersenne primes; [33] GIMPS has found all known Mersenne primes beginning with the 35th. #
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
Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm is O(k log 2 n log log n) = Õ(k log 2 n), where k is the number of times we test a random a, and n is the value we want to test for primality; see Miller–Rabin primality test for details.
The parameter k determines the accuracy of the test. The greater the number of rounds, the more accurate the result. [6] Input #1: n > 2, an odd integer to be tested for primality Input #2: k, the number of rounds of testing to perform Output: “composite” if n is found to be composite, “probably prime” otherwise
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A prime sieve or prime number sieve is a fast type of algorithm for finding primes. There are many prime sieves. The simple sieve of Eratosthenes (250s BCE), the sieve of Sundaram (1934), the still faster but more complicated sieve of Atkin [1] (2003), sieve of Pritchard (1979), and various wheel sieves [2] are most common.