Goldbach's comet
Richard Tobin

This graph (sometimes known as "Goldbach's Comet") shows the number of ways in which each even number up to a million can be expressed as the sum of two primes. For Goldbach's conjecture to be false, there must be a zero value somewhere off to the right.

A striking feature of the graph is the division into bands. Let's see why they appear.

The graph below is similar, but the points are colour-coded to indicate the small prime factors of the number. Multiples of 3 are green, multiples of 5 are red, and multiples of 7 are blue. These are combined so that multiples of 3 and 5 are yellow, multiples of 3 and 7 are cyan, multiples of 5 and 7 are magenta, and multiples of all three are light grey. Numbers that are a power of 2 times a prime are orange.

It's easy to see why small odd factors make such a difference. Consider this fragment of the set of pairs of odd numbers adding up to 100. We're looking for ones where the pair are both primes. Clearly any that are multiples of 3 — marked in green — are not prime, so any pair with a green number is ruled out:

45474951535557596163
55535149474543413937

Now consider the case for 102, which is a multiple of 3:

45474951535557596163
57555351494745434139

In the case of 102, only a third of the pairs are ruled out by having a multiple of 3, while for 100 two thirds were. Larger prime factors will have a similar but less pronounced effect.

We might therefore expect that if we take the pairs for a multiple of 3, and remove half of them — so that two thirds of the pairs have been removed instead of one third — then we would get a value similar for that for a non-multiple of 3. And indeed if for each distinct prime factor p>2, we multiply the number of pairs by (p-2)/(p-1), we get this graph:

The band structure has disappeared, as this close-up makes clear: