A study of the gaps between consecutive prime numbers

We investigate the sequence of gaps between consecutive prime numbers, by recasting Eratosthenes sieve as a recursion explicitly on the gaps. This approach preserves subsequences of gaps, known as constellations. We can count the exact numbers of constellations at each stage of Eratosthenes sieve.

For a constellation of a gap of 2 followed by a gap of 4, we simply concatenate the single digits as 24. This constellation occurs for example between the primes 5,7,11 and again between 11,13,17. The sequence of primes and their gaps begin as follows:

primes  2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79 ...
gaps      1  2  2  4   2   4   2   4   6   2   6   4   2   4   6   6   2   6   4   2   6 ...

Traditional approaches based on Hardy and Littlewood's seminal 1923 paper go immediately to probabilistic estimates of the frequencies with which particular gaps occur between primes. These estimates are very good, but they apply primarily to single gaps and to simple constellations (such as 24 or 242). The recursion extends the deterministic analysis of constellations, and allows us to apply probabilistic methods to constellations of any complexity. These probabilistic estimates agree well with the traditional ones, but also apply to situations where the traditional methods do not.

Why is this useful?

• The recursion works explicitly on the gaps.
• A lot of structure is preserved from one cycle of gaps to the next.
• All the gaps up to the square of the next prime are actually gaps between primes.
• Each possible addition between adjacent gaps occurs exactly once in the next stage of the recursion.
• Small constellations survive the recursion, so we can enumerate exactly how many copies of this constellation will occur in every subsequent cycle of gaps.