On nonconvex constellations among primes I
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913044618018816 |
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| author | Holt, Fred B. |
| author_facet | Holt, Fred B. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25896 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On nonconvex constellations among primes I Holt, Fred B. Number Theory 11N05, 11A41, 11B75 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. |
| title | On nonconvex constellations among primes I |
| topic | Number Theory 11N05, 11A41, 11B75 |
| url | https://arxiv.org/abs/2603.25896 |