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On nonconvex constellations among primes I

Fred B. Holt

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.

On nonconvex constellations among primes I

Abstract

Extending our work on the -tuple conjecture, we apply those methods to the Engelsma counterexamples (narrow constellations) of length and span . We track the evolution of these counterexamples from inadmissible driving terms starting in the cycle of gaps up through their first appearance in . We continue developing primorial coordinates for each admissible instance through a breadth-first exhaustive search through , 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 -counterexamples occur before . For each of the Engelsma -counterexamples we calculate its asymptotic relative population, among other constellations of length , and we study how these counterexamples work.

Paper Structure

This paper contains 9 sections, 3 theorems, 17 equations, 7 figures, 6 tables.

Table of Contents

  1. Introduction
  2. How do these counterexamples work?
  3. Indexing and scoring narrow admissible $k$-tuples
  4. Engelsma $(459,3242)$-counterexamples
  5. Why length $J=459$?
  6. Prefixes for the primorial coordinates
  7. Relative populations of $(459,3242)$-counterexamples.
  8. Searching for surviving instances
  9. Conclusion

Key Result

Lemma 1

From the Engelsma data, the smallest counterexamples to the convexity conjecture would occur for $({J,|s|)=(458,3240)}$ and ${(459,3242)}$.

Figures (7)

  • Figure 1: Convexity and nonconvexity. We are comparing the density of primes in an interval away from the origin, with the density of an interval of equal length starting at the origin.
  • Figure 2: One constellation in $(J,|s|)=(459,3242)$ plotted (in blue) as a segment of $\Delta \Phi(x,\mu)$, compared to the gaps between primes (in red) from $0$ to $n$. The small gaps in $\pi(x)$ create a huge early lead, but as larger gaps occur between primes, it gives narrow constellations an opportunity to overtake it. The inset shows where the graph for $s_4$ surpasses the sequence of prime gaps. We see that $s_4$ first crosses above the red graph for $(458,3240)$, and stays above for $(459,3242)$.
  • Figure 3: A section of Engelsma's table, shown at right, supplemented with columns for $J = k-1$, span $|s|=w-1$, and the true nonconvexity score $J-\pi(|s|)$. We list the primes $p_J$ in the second column for reference. The red segments show where in the sequence of primes the entries $|s|$ fall, illustrating $\pi(|s|)$.
  • Figure 4: One example $s_4$ of the $58$ narrow constellations in $(J,|s|)=(459,3242)$.
  • Figure 5: Listed here are the primorial coordinates for the unique instances of each of the 29 $(459,3242)$-counterexamples that have initial generator $\gamma_0=107$ in ${\mathcal{G}}(11^\#)$. The driving terms in ${\mathcal{G}}(p^\#)$ for $p < 113$ are inadmissible -- they do not survive. The counterexamples themselves appear in ${\mathcal{G}}(113^\#)$. The row of primorial coordinates for the prefix for our example $s_4$ is highlighted.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof