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This question was posed today by a student of mine, and I have not been able to find any relevant references.

Let $p_1, p_2, p_3, \ldots$ be the sequence of prime numbers listed in increasing order, and let $d_k = p_{k+1}-p_k$ be the $k$'th prime gap. Pick a large number $N$.

First, we count the number of indices $k$ between 1 and $N$ such that $d_k = 2$. Denote this number by $f(N)$. It is the number of pairs of twin primes found among the first $N+1$ primes.

Secondly, we count the number of indices $k$ between 1 and $N$ such that $d_k = 4$. Denote this number by $g(N)$.

Numerical evidence suggests that the ratio $f(N) / g(N)$ tends to 1 as $N$ tends to infinity. Are there any heuristic reasons to believe that the limit exists, and that it really is equal to 1? Is this a known conjecture, and if so, what is a good reference?

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    $\begingroup$ This is a case of the first Hardy-Littlewood conjecture on the asymptotic densities of constellations of primes. $\endgroup$
    – Douglas Zare
    Commented Apr 28, 2017 at 15:51
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    $\begingroup$ Your student might be interested in the recent discovery by Lemke Oliver and Soundararajan of somewhat surprising biases in prime gaps that come as consequences of the Hardy-Littlewood conjecture, see e.g. the popular article quantamagazine.org/… or my blog post terrytao.wordpress.com/2016/03/14/… $\endgroup$
    – Terry Tao
    Commented Apr 28, 2017 at 16:18
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    $\begingroup$ Think mod 6. A prime gap of 2 can only occur between $6k-1, 6k+1$ for some integer $k$; a prime gap of 4 can only occur between $6k+1$ and $6k+5$ for some integer $k$. $\endgroup$
    – Michael Lugo
    Commented Apr 28, 2017 at 17:03
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    $\begingroup$ @MichaelLugo What exactly does that establish? $\endgroup$
    – Wojowu
    Commented Apr 28, 2017 at 20:02

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The comments below the question point at conjectures and briefly at a heuristic suggesting the ratio tends to one. Although not a proof, the idea below gives more evidence for believing in there being about the same number of both sizes of gaps.

Start by doing a wheel sieve, but with two differences: record the differences between members, and extend the sequences out to infinity. I illustrate first with a few examples.

2 goes to 4 2 goes to 6 4 2 4 2 4 6 2 goes to 10 2 4 2 4 6 2 ...(33 more terms)... 2 6 4 2 4 2 10 2,

which patterns we now repeat (the first three):

2 2 2 2 2 2 2 2 2 2 2 ...

4 2 4 2 4 2 4 2 4 2 4 2 ...

6 4 2 4 2 4 6 2 6 4 2 ...

The repeating sequences can be viewed as gaps between numbers starting at 1 which are relatively prime to a primorial (2,6,30,210, etc.) There is a website primegaps.com maintained by Fred Holt which explores these sequences and does some statistical analysis. Upon request, I can deliver some AWK code to derive one sequence from the previous sequence, but you might enjoy building your own program.

The upshot is that one can prove that the number of occurrences of 4's and 2's is asymptotically the same for large enough portions of the sequence. The downside is that these portions are very long, and differ greatly from the sequence of prime gaps. However, one can analyze how the number of 2's varies from the number of 4's in parts of the sequence. For smaller parts, one finds little difference between the numbers, although for very small parts there are differences free of 2's and others free of 4's.

Gerhard "Free Numbers Free Of Numbers" Paseman, 2018.04.13.

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