Population models for gaps and constellations

Let p be a prime and q the next prime.

The recursion from the cycle of gaps G(p#) to the next cycle G(q#) -- in which q copies of G(p#) are concatenated and then certain adjacent gaps are fused -- is a discrete dynamic system.

Two observations enable us to model exact populations of gaps across the stages of this system:

1st Observation. The minimum span between two fusions is 2q.

This is because the fusions occur at the running sums indicated by the element-wise product q*G(p#),

and the minimum element in G(p#) is 2.

2nd Observation. Each possible fusion in G(p#) occurs exactly once in forming G(q#).

This is a result of the Chinese Remainder Theorem in this setting.

For each gap g, we define the driving terms for g to be those constellations (sequences of adjacent gaps) that under fusions will produce shorter driving terms for g or the gap g itself.

We sort the driving terms by their length j.

Shown here are the systems of driving terms for the gaps g from 2 through 12. We see most of these driving terms in G(5#).

If a gap g in G(p#) is less than 2q,

then for each instance of a driving term s for a gap g of length j in G(p#),

the concatenations of G(p#) create q copies of this instance of s.

Of the j+1 fusions in s, the j-1 interior fusions create driving terms for g of length j-1; the 2 exterior fusions remove those two copies as driving terms for g;

and q-j-1 copies of this instance of s survive intact in G(q#).

The populations and relative populations are modeled exactly by linear systems

Let p be a prime and q be the next larger prime.

For any gap g < 2q, the population of g and its driving terms in G(q#) can be calculated from their populations in G(p#) by the linear system

The system matrix is a banded matrix, and the size J should be (at least) the length of the longest driving term for the gap g. Note that the populations of gaps ultimately grow by factors of (q-2) over all the primes q > p.

All gaps are subject to the same population model, and the system matrix depends on the prime q and not on the gap g.

To complete the population models, we need to pick some prime p for which we can get the initial conditions (the initial populations) from G(p#).

Initial conditions

We take our first sample of initial conditions from

G(5#) = 6 4 2 4 2 4 6 2

With these initial conditions we can develop exact population models for the gaps g < 14, since 7 is the next prime after 5.

For gaps g=14 and larger, multiple fusions could occur in a single copy of a driving term, throwing off our counts.

For simplicity in comparing population models across gaps, we want to use the same starting cycle G(p#), and in order to model as many gaps as we can, we want p to be as large as we can make it. The challenge to increasing p is that the length of the cycle increases by factors of (p-1).

For our studies, we use G(37#).

A heat map of the initial conditions for G(37#) by gap g and length j is shown at left.

This heat map shows the populations for gaps up to g=2310 and lengths up to J=369.

2310 =11# is the span of one cycle of G(11#).

Also shown are the footprints of G(5#), G(7#), and G(11#)

across this range of gaps. We can see how the driving terms for larger gaps diminish in length as the sieve continues.

Taking initial conditions from G(37#), we can create exact population models for gaps up to g=80.

Relative population models

In this normalized system, the normalized populations w of the gaps g=2 and g=4 are both identically 1 in all cycles G(p#) with p > 2.

For all other gaps g, their normalized populations in G(q#) can be interpreted as their relative populations compared to the number of gaps g=2 or g=4.

Even though the system matrix M(q#) changes with each prime q, these have a beautifully simple eigenstructure. The matrix of left eigenvectors is an upper triangular Pascal matrix. The matrix of right eigenvectors is an upper triangular Pascal matrix of alternating sign. And the matrix of eigenvalues is diagonal. The eigenvalues are real, positive, and decreasing, with dominant eigenvalue 1.

The population of a gap g grows superexponentially by factors of (q-2) across stages of Eratosthenes sieve.

We normalize the linear system by dividing by (q-2).