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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.
The algorithm can be written in pseudocode as follows: algorithm lucas_primality_test is input: n > 2, an odd integer to be tested for primality. k, a parameter that determines the accuracy of the test. output: prime if n is prime, otherwise composite or possibly composite. determine the prime factors of n−1.
The algorithm can be written as follows: Inputs: n: a value to test for primality, n>3; k: a parameter that determines the number of times to test for primality Output: composite if n is composite, otherwise probably prime Repeat k times: Pick a randomly in the range [2, n − 2]
An example of a decision problem is deciding with the help of an algorithm whether a given natural number is prime. Another example is the problem, "given two numbers x and y, does x evenly divide y?" A method for solving a decision problem, given in the form of an algorithm, is called a decision procedure for that problem.
The following is pseudocode which combines Atkin's algorithms 3.1, 3.2, and 3.3 [1] by using a combined set s of all the numbers modulo 60 excluding those which are multiples of the prime numbers 2, 3, and 5, as per the algorithms, for a straightforward version of the algorithm that supports optional bit-packing of the wheel; although not specifically mentioned in the referenced paper, this ...
Optionally, perform trial division to check if n is divisible by a small prime number less than some convenient limit. Perform a base 2 strong probable prime test. If n is not a strong probable prime base 2, then n is composite; quit. Find the first D in the sequence 5, −7, 9, −11, 13, −15, ... for which the Jacobi symbol (D/n) is −1.
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AKS is the first primality-proving algorithm to be simultaneously general, polynomial-time, deterministic, and unconditionally correct. Previous algorithms had been developed for centuries and achieved three of these properties at most, but not all four. The AKS algorithm can be used to verify the primality of any general number given. Many ...