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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.
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
The Lucas–Lehmer–Riesel test is a particular case of group-order primality testing; we demonstrate that some number is prime by showing that some group has the order that it would have were that number prime, and we do this by finding an element of that group of precisely the right order.
A Lucas pseudoprime for a given (P, Q) pair is a positive composite integer n for which equation is true (see, [1] page 1391). A Lucas probable prime test is most useful if D is chosen such that the Jacobi symbol () is −1 (see pages 1401–1409 of, [1] page 1024 of, [5] or pages 266–269 of [2]).
The Baillie–PSW primality test is a probabilistic primality test that combines a Fermat or Miller–Rabin test with a Lucas probable prime test to get a primality test that has no known counterexamples. That is, there are no known composite n for which this test reports that n is probably prime.
To test the primality of a number, the GIMPS programs run a probable prime test, and, if the test produces a successful result, the number is almost conclusively a new prime number.
These numbers have been proved prime by computer with a primality test for their form, for example the Lucas–Lehmer primality test for Mersenne numbers. “!” is the factorial, “#” is the primorial, and () is the third cyclotomic polynomial, defined as + +.
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