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Certain number-theoretic methods exist for testing whether a number is prime, such as the Lucas test and Proth's test. These tests typically require factorization of n + 1, n − 1, or a similar quantity, which means that they are not useful for general-purpose primality testing, but they are often quite powerful when the tested number n is ...
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.
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
Input #1: b, the number of bits of the result Input #2: k, the number of rounds of testing to perform Output: a strong probable prime n while True: pick a random odd integer n in the range [2 b−1, 2 b −1] if the Miller–Rabin test with inputs n and k returns “probably prime” then return n
Suppose we wish to determine whether n = 221 is prime.Randomly pick 1 < a < 220, say a = 38.We check the above congruence and find that it holds: = (). Either 221 is prime, or 38 is a Fermat liar, so we take another a, say 24:
The Solovay–Strassen primality test, developed by Robert M. Solovay and Volker Strassen in 1977, is a probabilistic primality test to determine if a number is composite or probably prime. The idea behind the test was discovered by M. M. Artjuhov in 1967 [ 1 ] (see Theorem E in the paper).
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Assume that p − 1, where p is the smallest prime factor of n, can be modelled as a random number of size less than √ n. By Dixon's theorem, the probability that the largest factor of such a number is less than (p − 1) 1/ε is roughly ε −ε; so there is a probability of about 3 −3 = 1/27 that a B value of n 1/6 will yield a factorisation.