Surviving the sieve
2
Now we address the problem of survival. The gaps between primes are exactly the gaps that survive the fusions in the recursive construction across the cycles of gaps G(q#).
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The horizon of survival
Let p be a prime and q be the next larger prime.
In the cycle G(p#) the first gap is (q-1). The second gap is the gap g between q and the next prime.
The first fusion will be after the first gap, confirming q as a prime and g as a gap between primes.
The next fusion will be at the gap in G(p#) corresponding to q .
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In G(p#), all of the gaps from q up to q are confirmed as gaps between primes (except the last gap, which gets fused upon the elimination of q as a candidate prime).
For the cycle G(p#) we call the interval [q,q ] the horizon of survival.
2
2
2
In G(7#), all of the gaps (except the last one) up to 121 survive as gaps between primes
G(7#) = 10, 2 4 2 4 6 2 6 4 2 4 6 6 2 6 4 2 6 4 6 8 4 2 4 2 4 8 6 4 6 2 4 6 2 6 6 4 2 4 6 2 6 4 2 4 2, 10, 2
+
121
The horizon for survival for G(7#)
+
11
At the next stage of the sieve, the horizon of survival extends again but there is significant overlap with the previous horizon of survival.
In G(11#), all of the gaps (except the last one) up to 169 survive as gaps between primes
G(11#) = 12, 4 2 4 6 2 6 4 2 4 6 6 2 6 4 2 6 4 6 8 4 2 4 2 4 14 4 6 2 10 2 6 6 4 2 4 6 2, 10, 2 4 2, 12, 10, 2 4...
+
169
+
13
The horizon for survival for G(11#)
14 4 6 2 10 2 6 6 4
The new interval for survival added in G(11#), beyond the interval for survival from G(7#).
Again let p and q be consecutive primes, with the gap g=q - p. As the horizon for survival expands from p to q the added interval of survival is
2
2
q - p = (p+g) - p = 2pg + g = g (2p + g)
2
2
2
2
2
For samples over a short range of primes p, these intervals vary primarily by the sizes of the gaps g.
Sampling intervals of survival
We sample the intervals of survival over a range of primes. We count the numbers of the different sizes of gaps within each interval, color code these by the gap g = q - p, and create histograms of these counts.
We note that these samples appear to be fair samples from the corresponding cycles of gaps in two ways.
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First, within each column, that is for each size of gap, the counts across the different primes appear to be proportional to the gap g. The number of 2's, 4's, or 6's accumulated when the gap g=18 appears three times as large as when the gap g=6.
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Second, the relative heights across the columns reflect the relative populations given by the models for the value p.
Shown here are the relative population models, and we have marked the sections for p=3853 and p=14797 for comparison to the histograms of surviving gaps above.
The models are shown for those gaps up to g=80 whose prime factors are only 2, 3, and/or 5.
