Finding Large Primes

In this paper we present and expand upon procedures for obtaining a large k-digit prime number to an arbitrarily high probability. We use a layered approach. The first step is to limit the pool of random numbers to exclude numbers that are obviously composite. We remove numbers not ending in 1, 3, 7, or 9, then exclude numbers with a digital root of 3, 6, or 9. This sharply increases the probability of the random number being prime. We then use the prime number theorem to find the probability that a selected number eta is prime and use the Miller-Rabin test to increase the probability that eta is prime to an arbitrarily high degree. Conditional probabilities are computed and confirmed experimentally using the GNU GMP library.