A Generalized Hawkins Sieve and Prime $K$-tuplets

Abstract

The Hawkins random sieve, obtained from a simple probabilistic variation of Eratoshenes's sieve, provides a compelling model for the primes.Building on the Hawkins' sieve, we introduce a general random sieve, and prove analogs of both the Prime Number Theorem and Mertens' theorem.Applications include a new probabilistic model for prime k-tuplets. Introduction.1.1 Purpose.In this paper, we introduce a natural generalization of the Hawkins' random sieve, and prove analogs of both the Prime Number Theorem (PNT) and Mertens' theorem in the more general setting.As an application we present a new probabilistic model for prime k-tuplets.1.2 Background.When faced with the complexity of prime distribution theory, it is tempting to employ mathematical models.One of the most compelling models for the prime numbers is known as the Hawkins' primes.The Hawkins' model, first introduced by David Hawkins [13], is based on a simple stochastic variation of the sieve of Eratosthenes.Hawkins' sieve works as follows: Starting with all natural numbers two and larger, we identify X 1 = 2 as our first 'sieving number.'In the first step we independently sieve numbers from our list with probability 1/X 1 , and identify X 2 as the smallest surviving number which is larger than X 1 .In the second step, we sieve numbers from our remaining list with probability 1/X 2 and identify X 3 as the smallest surviving number which is larger than X 2 .If we carry on with the process, we produce a list {X 1 , X 2 , . . ., } of sieving numbers which are called Hawkins' primes.

Locations

• Rocky Mountain Journal of Mathematics - View - PDF