Pocklington Primality Test

High Quality Content by WIKIPEDIA articles! In mathematics, the Pocklington-Lehmer primality test is a primality test devised by Henry Cabourn Pocklington and Derrick Henry Lehmer to decide whether a given number N is prime. The output of the test is a proof that the number is prime or that primality could not be established. The test is simple once the theorem above is established. Given N, seek to find suitable a and q. If they can be obtained, then N is prime. Moreover, a and q are the certificate of primality. They can be quickly verified to satisfy the conditions of the theorem, confirming N as prime. A problem which arises is the ability to find a suitable q, that must satisfy (1) , (2) and be provably prime. It is even quite possible that such a q does not exist. This is a large probability, indeed only 57.8% of the odd primes, N, N le 10, 000 have such a q. To find a is not nearly so difficult. If N is prime, and a suitable q is found, each choice of a where 1 le a < N will satisfy a^{N-1} equiv 1pmod{N}, and so will satisfy (2) as long as ord(a) does not divide (N ¿ 1) / q.