Search results
Results from the WOW.Com Content Network
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.
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 ...
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
Libgcrypt uses a similar process with base 2 for the Fermat test, but OpenSSL does not. In practice with most big number libraries such as GMP, the Fermat test is not noticeably faster than a Miller–Rabin test, and can be slower for many inputs. [4] As an exception, OpenPFGW uses only the Fermat test for probable prime testing.
These numbers have been proved prime by computer with a primality test for their form, for example the Lucas–Lehmer primality test for Mersenne numbers. “!” is the factorial, “#” is the primorial, and () is the third cyclotomic polynomial, defined as + +.
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.
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.
In computational number theory, the Adleman–Pomerance–Rumely primality test is an algorithm for determining whether a number is prime. Unlike other, more efficient algorithms for this purpose, it avoids the use of random numbers, so it is a deterministic primality test .