On nonconvex constellations among primes I

TL;DR

This paper extends previous work on prime constellations, analyzing specific counterexamples and their evolution through prime cycles, with extensive computational results up to very large numbers.

Contribution

It develops methods to track prime constellations through multiple prime cycles and computes their properties, providing new insights into their distribution and rarity.

Findings

None of the counterexamples occur before 9.7×10^73.

Calculated asymptotic relative populations of counterexamples.

Extended calculations to include all terms in primorial expansion for the smallest generator.

Abstract

Extending our work on the $k$-tuple conjecture, we apply those methods to the Engelsma counterexamples (narrow constellations) of length $J=459$ and span $|s|=3242$. We track the evolution of these $58$ 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 ${\mathcal G}(211^\#)$, at which point we need to develop strategies for depth-first searches for an instance that would survive Eratosthenes sieve. Our calculations show that {\em none} of the $(459,3242)$-counterexamples occur before $9.7\,E73$. For each of the $58$ Engelsma $(459,3242)$-counterexamples we calculate its asymptotic relative population, among other constellations of length $J=459$, and we study how these counterexamples work. In this version (9 April) we have completed the calculations in Table 6 to include all of the terms in primorial expansion for the smallest initial generator and corrected a typographical error on page 6.