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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.
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
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]
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
While Carmichael numbers are substantially rarer than prime numbers (Erdös' upper bound for the number of Carmichael numbers [3] is lower than the prime number function n/log(n)) there are enough of them that Fermat's primality test is not often used in the above form.
The isPrime function was inaccurate, as range doesn't include the higher end, so e.g. if checking for primality of 9, it would try numbers from 2 to 2, and conclude it was prime. I've added 1 to the upper end of the range so that the isPrime function works, in case anyone else comes along and tries to use it.
where ⌊ x ⌋ is the floor function, which denotes the greatest integer less than or equal to x and the p i run over all primes ≤ √ x. [1] [2] Since the evaluation of this sum formula becomes more and more complex and confusing for large x, Meissel tried to simplify the counting of the numbers in the Sieve of Eratosthenes. He and Lehmer ...