On nonconvex constellations among primes II: (458,3240)

TL;DR

This paper extends previous analysis of prime constellations related to the $k$-tuple conjecture, focusing on Engelsma counterexamples of specific lengths and spans, and tracks their evolution through prime cycles.

Contribution

It develops a detailed analysis of the evolution and population of specific prime constellations, extending prior work to new counterexamples and cycles.

Findings

No $(458,3240)$ constellation occurs outside its parent until cycle ${ m G}(227^ig#)$

Early evolution of constellations is dominated by their parent cycles

Calculates asymptotic relative populations of counterexamples

Abstract

Extending our work on the $k$-tuple conjecture, we previously applied those methods to the Engelsma counterexamples (narrow constellations) of length $J=459$ and span $|s|=3242$. Here we extend that analysis to the $116$ Engelsma counterexamples of length $J=458$ and $|s|=3240$. We track the evolution of these $116$ counterexamples from inadmissible driving terms starting in the cycle of gaps ${\mathcal G}(11^\#)$ up through their first appearance in ${\mathcal G}(113^\#)$. We continue developing primorial coordinates for each admissible instance through a breadth-first exhaustive search through ${\mathcal G}(211^\#)$. Each of the $(458,3240)$ constellations sits inside a $(459,3242)$ constellation, which we call its {\em parent}. We show that no $(458,3240)$ constellation occurs outside of its parent until the cycle ${\mathcal G}(227^\#)$. The early evolution of the $(458,3240)$ constellations is dominated by the evolution of their parents, which we have previously studied. For each $(458,3240)$-counterexample we calculate its asymptotic relative population, among other constellations of length $J=458$.