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The constellations at the middle of the cycle of gaps

Under the recursion, in addition to observing structure in the cycle as a whole, we can identify structure in the middle of the cycle.

Recall that except for the final gap of 2, the cycle of gaps G(Pk#) is symmetric. In the middle of the cycle G(Pk#) there is the constellation

... 2j 2j-1 ... 8 4 2 4 2 4 8 ... 2j-1 2j ...


in which j is the largest integer such that 2j < Pk+1.


Why? We consider reducibility and irreducibility relative to Pk#, the product of the primes from 2 through Pk. Reducible numbers in Z mod Pk# are known to be composite, and the irreducible ones are generators for this group and possible primes. That is, if they do have nontrivial prime factors, each of those factors is greater than Pk.

Many of these observations are applications of the divisibility rule: if p divides a+b and p divides a, then p divides b.

In Z mod Pk#, the midpoint is mk = Pk# /2 which is the product of the odd primes from 3 through Pk. Clearly mk is reducible, and since mk-1 and mk+1 are both even, these are also reducible. In contrast, mk-2 and mk+2 are both irreducible. So the middle gap is the 4 in between these two generators.

For all i such that 2i < mk, the numbers mk+2i and mk-2i are generators in Z mod Pk#. From mk to mk+2Pk+1, the numbers mk+2, mk+4, mk+8, ..., mk+2j+1 are the only generators. All the numbers in between are reducible. So after the central gap of 4, which gets us from mk-2 to mk+2, we have the constellation

... 4 2 4 8 ... 2j

 

The middle of the cycle, under recursion

What happens to the middle of the cycle of gaps in the next stage of the recursion? Under the concatenation in the second step, a copy of the middle of the cycle again ends up in the middle of the new cycle under construction. In the third step, the additions closest to the center of the cycle come from Pk+1 times the gap 4 at the middle of the cycle. These two additions remove mk+1-2Pk+1 and mk+1+2Pk+1.

Beyond the central 4 and the sequence of powers of two, the middle of the cycle of gaps corresponds to the numbers mk+2Pk+i and mk+4Pk+i interspersed with the numbers mk+2i. For example, consider the middle of

G(7#) = ... 8 4 2 4 2 4 8 6 4 6 2 4 6 2 6 6 4 2 ...

 

The next prime is 11, and following the central gap 4 we see the constellation of powers of two 2 4 8. With half the central 4, these gaps sum to 16. The next gap 6 takes the sum to 22 = 2*11. This gap 6 is part of a constellation 6 4 6 which sums to 16 (the next power of two). While that 6 brought the sum to 22, the 4 brings the sum to 26 = 2*13, and then the next 6 brings the sum to the power of two 32.

By G(13#) the constellation 6 4 6 has been summed to the single gap 16:


G(13#) = ... 16 8 4 2 4 2 4 8 16 2 4 8 12 4 2 4 6 2 6 4 6 2 12 10 2 4 2 4 ...

 

Following the central gap 4 we have the constellation 2 4 8 16, and then the constellation 2 4 8 12 4 2, which sums to 32. In this constellation, the initial 2 brings the running sum to 34 = 2*17. Then we have a sequence of twice the gaps between subsequent primes. That is, G(13#) starts with the constellation 16 2 4 6 2 6 ..., and we see the first few gaps among subsequent primes, 2 4 6 2, show up doubled 4 8 12 4 near the middle of the cycle.