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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.

Our papers related to this work

On Polignac's Conjecture - 9 Feb 2014, Fred B. Holt, Helgi Rudd

By generalizing our methods, we are able to demonstrate that for every even number 2n the gap g=2n occurs infinitely often through the stages of Eratosthenes sieve. Moreover, we show that asymptotically the ratio of the number of gaps g=2n to the number of gaps g=2 at each stage of Eratosthenes sieve converges to the estimates made for gaps among primes by Hardy and Littlewood in Conjecture B of their 1923 paper.

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On small gaps among primes - 29 Dec 2013, Fred B. Holt, Helgi Rudd

In this paper, we study these systems of constellations of a fixed sum. Viewing them as discrete dynamic systems, we are able to characterize the populations of constellations for sums including the first few primorial numbers: 2, 6, 30. Since the eigenvectors of the discrete dynamic system are independent of the prime -- that is, independent of the stage of the sieve -- we can characterize the asymptotic behavior exactly. In this way we can give exact ratios of the occurrences of the gap 2 to the occurrences of other small gaps for all stages of Eratosthenes sieve.

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Estimating constellations among primes - I. Uniformity - 8 Dec 2013, Fred B. Holt, Helgi Rudd

A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which are known as constellations. In this paper, we use those enumerations to estimate the numbers of these constellations that occur as constellations among prime numbers, and we compare these estimates with computational results. We include in our estimates the constellations corresponding to three and four consecutive primes in arithmetic progression. For these initial estimates, we assume that the copies of a given constellation tend toward a uniform distribution in the cycle of gaps, as the recursion progresses. Our simple estimates based on the recursion of gaps and the assumption of uniformity appear to have correct asymptotic behavior, and they exhibit a systematic error correlated to length of the constellation.

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Expected gaps between prime numbers - 6 Jun 2007, Fred B. Holt

We study the gaps between consecutive prime numbers directly through Eratosthenes sieve. Using elementary methods, we identify a recursive relation for these gaps and for specific sequences of consecutive gaps, known as constellations. Using this recursion we can estimate the numbers of a gap or of a constellation that occur between a prime and its square. This recursion also has explicit implications for open questions about gaps between prime numbers, including three questions posed by Erdős and Turán.

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