Models for gaps $g=2p_1$

TL;DR

This paper develops a mathematical model for the distribution of prime gaps of size $g=2p_1$, extending previous models by incorporating initial conditions from cycles of gaps in Eratosthenes sieve, and demonstrates its application with specific examples.

Contribution

It introduces a method to produce models for prime gaps of size $g=2p_1$ using initial conditions from cycles of gaps, advancing the understanding of prime gap distributions.

Findings

Model for $g=2p_1$ derived from initial conditions.

Application example for gap $g=82$ using $ ext{G}(37^ ext{#})$.

Framework for extending models to larger gaps.

Abstract

We have shown previously that at each stage of Eratosthenes sieve there is a corresponding cycle of gaps $\mathcal{G}(p_0^\#)$. We can view these cycles of gaps as a discrete dynamic system, and from this system we can obtain exact models for the populations and relative populations of gaps $g < 2p_1$ if we can get the initial conditions from $\mathcal{G}(p_0^\#)$. In this addendum we have shown that we can produce the model for $g=2p_1$ from these initial conditions. This model requires one special iteration to track the count from $\mathcal{G}(p_0^\#)$ to $\mathcal{G}(p_1^\#)$, after which we can use the general model for these populations. As a specific example we exhibit the model for the gap $g=82$ using $\mathcal{G}(37^\#)$ for initial conditions. We show further that in order to produce the models for $g=2p_1+2$ and beyond from initial conditions in $\mathcal{G}(p_0^\#)$, we would have to track subpopulations of the driving terms until the general model applies, that is until $g < 2p_{k+1}$. This work serves as an addendum to the existing references "Patterns among the Primes" and "Combinatorics of the gaps between primes". We do not duplicate that background here, beyond summarizing a few needed results.