Twin primes are much beloved. But a computer search has shown that among numbers less than a trillion, most common distance between successive primes is 6. It seems this goes on for quite a while longer…

… but Andrew Odlyzko, Michael Rubinstein and Marek Wolf have persuaded most experts that somewhere around x = 1.7427 × 10 35 x = 1.7427 \times 10^{35} , the most common gap between consecutive primes less than x x switches from 6 to 30:

Andrew Odlyzko, Michael Rubinstein, and Marek Wolf, Jumping champions, Experimental Mathematics 8 (1999), 107–118.

This is a nice example of how you may need to explore very large numbers to understand the true behavior of primes.

They give a sophisticated heuristic argument for their claim—not a rigorous proof. But they also checked the basic idea using Maple’s ‘probable prime’ function. It takes work to check if a number is prime, but there’s a much faster way to check if it’s probably prime in a certain sense. Using this, they worked out the gaps between probable primes from 10 30 10^{30} and 10 30 + 10 7 10^{30}+10^7 . They found that there are 5278 gaps of size 6 and just 5060 of size 30. They also worked out the gaps between probable primes from 10 40 10^{40} and 10 40 + 10 7 10^{40}+10^7 . There were 3120 of size 6 and 3209 of size 30.

So, it seems that somewhere between 10 30 10^{30} and 10 40 10^{40} , the number 30 replaces 6 as the most probable gap between successive primes!

Using the same heuristic argument, they argue that somewhere around 10 450 10^{450} , the number 30 ceases to be the most probable gap. The number 210 replaces 30 as the champion—and reigns for an even longer time.

Furthermore, they argue that this pattern continues forever, with the main champions being the ‘primorials’:

2 2

2 ⋅ 3 = 6 2 \cdot 3 = 6

2 ⋅ 3 ⋅ 5 = 30 2 \cdot 3 \cdot 5 = 30

2 ⋅ 3 ⋅ 5 ⋅ 7 = 210 2 \cdot 3 \cdot 5 \cdot 7 = 210

2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11 = 2310 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 = 2310

etc.