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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 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
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),
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.
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 ...
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.
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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 .