Every even number above 2 seems to be the sum of two primes: 4 = 2 + 2, 28 = 5 + 23, 100 = 3 + 97. Computers have checked every even number up to 4 × 1018 without finding one that isn't, but nobody has proved that it always works.
Fold a number
28 = 5 + 23 = 11 + 17. Fold the line from 0 to 28 at 14 and every column adds up to 28, so a column with a prime at both ends is a way.
Goldbach's comet
Each dot is an even number, placed by how many ways it splits into two primes. Multiples of 3 make the upper band: they get about twice as many ways as the rest. Point at a dot to see it, and tap it to fold it.
Your browser counts the ways for every even number from 4 to 100,000, left to right.
Multiples of 3
Other even numbers
Fewest ways: 4, 6, 8 and 12, with one each. Most: 99,330, with 2,168.
Why multiples of 3 do better
Every prime above 3 is one more or one less than a multiple of 3. Take an even number that isn't a multiple of 3, like 100. About half the primes are the wrong kind for it: 100 − 7 = 93, a multiple of 3, so it can't be prime. A multiple of 3, like 102, has no wrong kind, because either kind of prime leaves the other kind behind. So multiples of 3 get about twice the chances, and on the right half of the comet they average twice as many ways.
5 and 7 do the same on a smaller scale. Multiples of 5 average a third more ways than the rest, and multiples of 7 a fifth more, which is why each band has fainter stripes inside it. G. H. Hardy and J. E. Littlewood turned this into a formula that predicts the comet's shape closely, but a prediction isn't a proof.
Christian Goldbach wrote to Leonhard Euler on 7 June 1742 with a guess about splitting numbers into primes; he counted 1 as a prime, as some people did then. In his reply, Euler recalled a simpler version Goldbach had once told him: every even number is a sum of two primes. He called it a theorem he was completely sure of, even though he couldn't prove it. Nobody has proved it since.
Nils Pipping checked every even number up to 100,000 by hand in 1938, the same range as the comet here. By 2012 a computer search led by Tomás Oliveira e Silva had checked every even number up to 4 × 1018. In 2013 Harald Helfgott published a proof of a weaker version for odd numbers: every odd number above 5 is a sum of three primes.
Small numbers have the least room. 4, 6, 8 and 12 have one way each, and 68 is the last with two or fewer: every even number from 70 to 100,000 has at least three.