Small gaps between primes

The usual sources of online science news seldom deal with important stories in mathematics. This is especially annoying when one considers the kind of trivia often reported in other scientific fields. So things are proceding true to form with respect to the news that Dan Goldston and colleagues appear to finally have a valid proof of their "prime gaps" theorem.

But the information is out there if you know to look for it and you look hard enough. The basics are in this note on MathForge. There's an interview with Dan Goldston in his hometown paper. And you can read the technical paper on Small gaps between primes or almost primes at ArXiv.org.

So what does the theorem actually say? It isn't that hard to explain. The prime number theorem states that the average gap between two consecutive primes p_n and p_n+1 is about log(p_n), where "log" is the natural logarithm. But that is a very rough estimate. It is suspected that the gap can be as small as possible (two) infinitely often -- a proposition known as the twin prime conjecture. This is one of the most famous (to mathematicians) unsolved problems in number theory, right up there with the Riemann Hypothesis and Goldbach's conjecture.

The result that Goldston and colleagues have sought to prove is a little weaker. Symbolically it is that

lim inf_n→∞ (p_n+1 - p_n)/log(p_n) = 0

What this says is that the ratio of the difference between the n+first prime and the n^th prime to the natural logarithm of the n^th prime is smaller than any positive number ε for infinitely many n, no matter how small ε is. Using the same notation, one could express the twin prime conjecture as

lim inf_n→∞ (p_n+1 - p_n) = 2

Exciting stuff, no? Well, it is to number theorists, anyhow.