# 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(P _{k}#)* is symmetric.
In the middle of the cycle

*G(P*there is the constellation

_{k}#)^{j}2

^{j-1}... 8 4 2

**4**2 4 8 ... 2

^{j-1}2

^{j}...

in which *j* is the largest integer such that * 2 ^{j} < P_{k+1}*.

**Why?** We consider reducibility and irreducibility relative to *P _{k}#*, the product of the primes from

*2*through

*P*. Reducible numbers in

_{k}*Z*mod

*P*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

_{k}#*P*.

_{k}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 *P _{k}#*, the midpoint is

*m*which is the product of the odd primes from

_{k}= P_{k}# /2*3*through

*P*. Clearly

_{k}*m*is reducible, and since

_{k}*m*and

_{k}-1*m*are both even, these are also reducible. In contrast,

_{k}+1*m*and

_{k}-2*m*are both irreducible. So the middle gap is the

_{k}+2*4*in between these two generators.

For all *i* such that *2 ^{i} < m_{k}*, the numbers

*m*and

_{k}+2^{i}*m*are generators in

_{k}-2^{i}*Z*mod

*P*. From

_{k}#*m*to

_{k}*m*, the numbers

_{k}+2P_{k+1}*m*,

_{k}+2*m*,

_{k}+4*m*, ...,

_{k}+8*m*are the only generators. All the numbers in between are reducible. So after the central gap of

_{k}+2^{j+1}*4*, which gets us from

*m*to

_{k}-2*m*, we have the constellation

_{k}+2*...*

**4**2 4 8 ... 2^{j}## 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 *P _{k+1}* times the gap

*4*at the middle of the cycle. These two additions remove

*m*and

_{k+1}-2P_{k+1}*m*.

_{k+1}+2P_{k+1} Beyond the central *4* and the sequence of powers of two, the middle of the cycle of gaps corresponds to
the numbers *m _{k}+2P_{k+i}* and

*m*interspersed with the numbers

_{k}+4P_{k+i}*m*. For example, consider the middle of

_{k}+2^{i}*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

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.