On nonconvex constellations among primes II: (458,3240)
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914579308609536 |
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| author | Holt, Fred B. |
| author_facet | Holt, Fred B. |
| contents | 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 $J=458$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19165 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On nonconvex constellations among primes II: (458,3240) Holt, Fred B. Number Theory 11N05, 11N13, 11B75 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 $J=458$. |
| title | On nonconvex constellations among primes II: (458,3240) |
| topic | Number Theory 11N05, 11N13, 11B75 |
| url | https://arxiv.org/abs/2605.19165 |