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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 ...
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).
Whether a value of x is composite is equivalent to of whether x is a member of COMPOSITE. It can be shown that COMPOSITE ∈ NP by verifying that it satisfies the above definition (if we identify natural numbers with their binary representations). COMPOSITE also happens to be in P, a fact demonstrated by the invention of the AKS primality test ...
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.
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] If (), then return composite
All prime numbers from 31 to 6,469,693,189 for free download. Lists of Primes at the Prime Pages. The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range. Interface to a list of the first 98 million primes (primes less than 2,000,000,000) Weisstein, Eric W. "Prime Number Sequences". MathWorld.
One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime (the factors need not be distinct, hence squares of primes are included). A composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to ...
Miller–Rabin primality test: a probabilistic algorithm for testing whether a given number n is prime or composite. If n is composite, the test will detect n as composite WHP. There is a small chance that we are unlucky and the test will think that n is prime.