Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim

Abstract

In early 2005, Dan Goldston, Janos Pintz, and Cem Yildirim [12] made a spectacular breakthrough in the study of prime numbers. Resolving a long-standing open problem, they proved that there are infinitely many primes for which the gap to the next prime is as small as we want compared to the average gap between consecutive primes. Before their work, it was known only that there were infinitely many gaps which were about a quarter the size of the average gap. The new result may be viewed as a step towards the famous twin prime conjecture that there are infinitely many prime pairs p and p+2, the gap here being 2, the smallest possible gap between primes. Perhaps most excitingly, their work reveals a connection between the distribution of primes in arithmetic progressions and small gaps between primes. Assuming certain (admittedly difficult) conjectures on the distribution of primes in arithmetic progressions, they are able to prove the existence of infinitely many prime pairs that differ by at most 16. The aim of this article is to explain some of the ideas involved in their work. Let us begin by explaining the main question in a little more detail. The number of primes up to x, denoted by π(x), is roughly x/ log x for large values of x; this is the celebrated Prime Number Theorem. Therefore, if we randomly choose an integer near x, then it has about a 1-in-log x chance of being prime. In other words, as we look at primes around size x, the average gap between consecutive primes is about log x. As x increases, the primes get sparser and the gap between consecutive primes tends to increase. Here are some natural questions about these gaps between prime numbers. Do the gaps always remain roughly about size log x, or do we sometimes get unexpectedly large gaps and sometimes surprisingly small gaps? Can we say something about the statistical distribution of these gaps? That is, can we quantify how often the gap is between, say, α log x and β log x, given 0 ≤ α < β? Except for the primes 2 and 3, clearly the gap between consecutive primes must be even. Does every even number occur infinitely often as a gap between consecutive primes? For example, the twin prime conjecture says that the

Locations

• arXiv (Cornell University) - View
• Bulletin of the American Mathematical Society - View - PDF