Finding Large Primes

TL;DR

This paper introduces a layered method for efficiently generating large prime numbers by filtering candidates and applying probabilistic primality tests, improving the likelihood of primality with high confidence.

Contribution

It presents a novel layered approach combining digital filtering and probabilistic tests to generate large primes with high probability efficiently.

Findings

Reduced candidate pool significantly lowers primality testing costs.

Combined use of Euler's and Miller-Rabin tests increases confidence in primality.

Implementation with GNU GMP library demonstrates practical effectiveness.

Abstract

In this paper we present and expand upon procedures for obtaining large d digit prime number to an arbitrary probability. We use a layered approach. The first step is to limit the pool of random number to exclude numbers that are obviously composite. We first remove any number ending in 1,3,7 or 9. We then exclude numbers whose digital root is not 3, 6, or 9. This sharply reduces the probability of the random number being composite. We then use the Prime Number Theorem to find the probability that the selected number n is prime and use primality tests to increase the probability to an arbitrarily high degree that n is prime. We apply primality tests including Euler's test based on Fermat Little theorem and the Miller-Rabin test. We computed these conditional probabilities and implemented it using the GNU GMP library.