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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.
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.
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. Thus, testing with 2, 3, and 5 suffices up to n = 48 not just 25 because the square of the next prime is 49, and below n = 25 just 2 and 3 are sufficient.
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
This occurs for example when n is a probable prime to base a but not a strong probable prime to base a. [20]: 1402 If x is a nontrivial square root of 1 modulo n, since x 2 ≡ 1 (mod n), we know that n divides x 2 − 1 = (x − 1)(x + 1); since x ≢ ±1 (mod n), we know that n does not divide x − 1 nor x + 1.
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]
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.