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Explicit bounds for the graphicality of the prime gap sequence

Keshav Aggarwal, Robin Frot, Haozhe Gou, Hui Wang

TL;DR

The paper establishes explicit unconditional thresholds for the graphicity of the first $n$ prime gap sequence, proving $PD_n$ is graphic for all $n\ge\exp\exp(30.5)$ and that, for $n\ge\exp\exp(34.5)$, every realization $G_n$ satisfies the DPG-graphic condition with respect to the next gap. It achieves this by combining a refined Erdős–Gallai criterion for graphicity with explicit zero-free regions and zero-density bounds for the Riemann zeta function, plus a Heath-Brown–type first-moment analysis of large prime gaps. The results hinge on sharpening the first-moment bounds for large gaps and on ensuring sufficiently large matchings via Vizing-type arguments in the DPG framework. These explicit bounds connect prime-gap distributions to graph-theoretic realizability, yielding concrete thresholds that enable an infinite Degree-Preserving Growth process on the prime-gap graph sequence.

Abstract

We establish explicit unconditional results on the graphic properties of the prime gap sequence. Let \( p_n \) denote the \( n \)-th prime number (with $p_0=1$) and \( \mathrm{PD}_n = (p_\ell - p_{\ell-1})_{\ell=1}^n \) be the sequence of the first \( n \) prime gaps. Building upon the recent work by Erdős \emph{et al}, which proved the graphic nature of \( \mathrm{PD}_n \) for large \( n \) unconditionally, and for all \( n \) under RH, we provide the first explicit unconditional threshold such that:\\ (1) For all \( n \geq \exp\exp(30.5) \), \( \mathrm{PD}_n \) is graphic.\\ (2) For all \( n \geq \exp\exp(34.5) \), every realization \( G_n \) of \( \mathrm{PD}_n \) satisfies that \( (G_n, p_{n+1}-p_n) \) is DPG-graphic. Our proofs utilize a more refined criterion for when a sequence is graphic, and better estimates for the first moment of large prime gaps proven through an explicit zero-free region and explicit zero-density estimate for the Riemann zeta function.

Explicit bounds for the graphicality of the prime gap sequence

TL;DR

The paper establishes explicit unconditional thresholds for the graphicity of the first prime gap sequence, proving is graphic for all and that, for , every realization satisfies the DPG-graphic condition with respect to the next gap. It achieves this by combining a refined Erdős–Gallai criterion for graphicity with explicit zero-free regions and zero-density bounds for the Riemann zeta function, plus a Heath-Brown–type first-moment analysis of large prime gaps. The results hinge on sharpening the first-moment bounds for large gaps and on ensuring sufficiently large matchings via Vizing-type arguments in the DPG framework. These explicit bounds connect prime-gap distributions to graph-theoretic realizability, yielding concrete thresholds that enable an infinite Degree-Preserving Growth process on the prime-gap graph sequence.

Abstract

We establish explicit unconditional results on the graphic properties of the prime gap sequence. Let denote the -th prime number (with ) and \( \mathrm{PD}_n = (p_\ell - p_{\ell-1})_{\ell=1}^n \) be the sequence of the first prime gaps. Building upon the recent work by Erdős \emph{et al}, which proved the graphic nature of for large unconditionally, and for all under RH, we provide the first explicit unconditional threshold such that:\\ (1) For all \( n \geq \exp\exp(30.5) \), is graphic.\\ (2) For all \( n \geq \exp\exp(34.5) \), every realization of satisfies that \( (G_n, p_{n+1}-p_n) \) is DPG-graphic. Our proofs utilize a more refined criterion for when a sequence is graphic, and better estimates for the first moment of large prime gaps proven through an explicit zero-free region and explicit zero-density estimate for the Riemann zeta function.
Paper Structure (12 sections, 13 theorems, 89 equations)

This paper contains 12 sections, 13 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. DPG-process
  3. Notations.
  4. Preliminaries
  5. Some explicit estimates
  6. Criteria for sequences being graphic
  7. Application of Vizing's theorem
  8. First moment of large prime gaps
  9. Proof of Theorem \ref{['MainThm']}
  10. Case 1: $k_N\geq \sqrt{p_n}+1$
  11. Case 2: $k_N\leq \sqrt{p_n}$
  12. About the DPG-process

Key Result

Theorem 1.3

One may take $n_0= \exp\exp(30.5)$. That is, for all $n \geq \exp\exp(30.5)$, the first $n$ prime gap sequence $\mathrm{PD}_n$ is graphic.

Theorems & Definitions (23)

  • Conjecture 1.1: Existence; Toroczkai, 2016
  • Conjecture 1.2: Structural; Toroczkai, 2016
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2: Delange Delange
  • Lemma 2.3: Erdős et al. EHKMMT
  • Lemma 2.4: Dudek Dudek
  • Remark 1
  • ...and 13 more