Primes and Probability: The Hawkins Random Sieve

While prime numbers are the fundamental building blocks of the integers, understanding how they are spread within the integers has turned out to be hard work. For example, the Prime Number Theorem stood as a conjecture for nearly a hundred years, and anyone who bags the Riemann Hypothesis first will be a million dollars richer. Modern cryptography assumes that the primes will retain their secrets for some time to come. In the presence of hard problems, it is tempting to employ models. A good model should provide an approximation of reality which is simple enough to understand, yet accurate enough to be useful. While there are several models for the primes, in this paper we tell the story of a beautiful and compelling probabilistic model known as the Hawkins primes. First introduced by David Hawkins in this MAGAZINE [11], the model is based on a simple variation of the sieve of Eratosthenes. Over the past fifty years, the Hawkins model has been used to predict the truth, in the strongest probabilistic sense, of results (both established and conjectured) concerning the distribution of the prime numbers, including the Twin Primes Conjecture and the Riemann Hypothesis. Also, the model (or generalizations thereof) has potential to shed light on interesting integer sequences other than the primes.