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A primality test is an algorithm for determining whether an input number is prime.Among other fields of mathematics, it is used for cryptography.Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not.
Fermat's little theorem states that if p is prime and a is not divisible by p, then a p − 1 ≡ 1 ( mod p ) . {\displaystyle a^{p-1}\equiv 1{\pmod {p}}.} If one wants to test whether p is prime, then we can pick random integers a not divisible by p and see whether the congruence holds.
A prime number is a natural number that has exactly two distinct natural number divisors: the number 1 and itself. To find all the prime numbers less than or equal to a given integer n by Eratosthenes' method: Create a list of consecutive integers from 2 through n: (2, 3, 4, ..., n). Initially, let p equal 2, the smallest prime number.
Check if n is a perfect power: if n = a b for integers a > 1 and b > 1, then output composite. Find the smallest r such that ord r (n) > (log 2 n) 2. If r and n are not coprime, then output composite. For all 2 ≤ a ≤ min (r, n−1), check that a does not divide n: If a|n for some 2 ≤ a ≤ min (r, n−1), then output composite.
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
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.
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A definite bound on the prime factors is possible. Suppose P i is the i 'th prime, so that P 1 = 2, P 2 = 3, P 3 = 5, etc. Then the last prime number worth testing as a possible factor of n is P i where P 2 i + 1 > n; equality here would mean that P i + 1 is a factor.