A short note on the computation of the generalised Jacobsthal function for paired progressions

TL;DR

This paper extends algorithms to compute the paired Jacobsthal function for primorial numbers, confirming the conjectured bounds up to primes 73, which has implications for Goldbach's and prime pairs conjectures.

Contribution

It introduces adapted algorithms for the paired Jacobsthal function and provides extensive computational verification up to primes 73.

Findings

All computed values satisfy the conjectured bounds.

The algorithms successfully handle primorial numbers for primes up to 73.

Results support the conjecture's applicability to Goldbach's and prime pairs conjectures.

Abstract

Jacobsthal's function was recently generalised for the case of paired progressions. It was proven that a specific bound of this function is sufficient for the truth of Goldbach's conjecture and of the prime pairs conjecture as well. We extended and adapted algorithms described for the computation of the common Jacobsthal function, and computed respective function values of the paired Jacobsthal function for primorial numbers for primes up to 73. All these values fulfil the conjectured specific bound. In addition to this note, we provide a detailed review of the algorithmic approaches and the complete computational results in ancillary files.

Tables

Table 1. Table 1: Results of computation.

$n$ $p_n$ $h_2\left(n\right)$ $p_n^2$-$p_n$          1 2 2 2        2 3 6 6        3 5 18 20        4 7 30 42        5 11 66 110        6 13 150 156        7 17 192 272        8 19 258 342        9 23 366 506        10 29 450 812        11 31 570 930

$n$ $p_n$ $h_2\left(n\right)$ $p_n^2$-$p_n$          12 37 708 1332        13 41 894 1640        14 43 1044 1806        15 47 1284 2162        16 53 1422 2756        17 59 1656 3422        18 61 1902 3660        19 67 2190 4422        20 71 2460 4970        21 73 2622 5256

Equations

$$j(n)=\min \{m\in \mathbb{N}\mid \forall a\in \mathbb{Z}\exists q\in \{1,\dots ,m\}:a+q\perp n\}.$$

$$⟨a,b{⟩}_m=\{(a+i,b+i)\in {\mathbb{Z}}^2\mid i=1,\dots ,m\}.$$

$$\begin{array}{rl}{j}_2(n)=\min \{m\in \mathbb{N}\mid & \forall (a,b)\in {\mathbb{Z}}^{2}with2\mid (b-a):\\ & \exists q\in \{1,\dots ,m\}:n\perp (a+q,b+q)\}.\end{array}$$

$$\begin{array}{rl}{h}_2(n)=\min \{m\in \mathbb{N}\mid & \forall (a,b)\in {\mathbb{Z}}^{2}with2\mid (b-a):\\ & \exists q\in \{1,\dots ,m\}\forall i\in \{1,\dots ,n\}:p_i\perp (a+q,b+q)\}.\end{array}$$

Full text

1 Introduction

Henceforth, we denote the set of integral numbers by $\mathbb{Z}$ and the set of naturalnumbers, i.e. positive integers, by $\mathbb{N}$. Let $\mathbb{P}=\{p_{i}\mid i\in\mathbb{N}\}$ be the set of prime numbers with $p_{1}=2$.

The commonly known Jacobsthal function $j(n)$ is defined to be the smallest natural number $m$, such that every sequence of $m$ consecutive integers contains at least one integer coprime to $n$ [2, 1].

Definition 1**.**

For $n\in\mathbb{N}$, the Jacobsthal function $j(n)$ is defined as

[TABLE]

Jacobshal remarked that the entire function is determined by its values forproducts of distinct primes [2]. The particular case of primorial numbers is thereforeof great interest. The function values at these points characterise the maximum growth of the function.

The Jacobsthal function was recently generalised for the case of paired progressions. A paired progression [4] is defined as an ordered sequence of consecutive pairs of integers of a definite length.

Definition 2**.**

Let $a,b\in\mathbb{Z}$ and $m\in\mathbb{N}$. A paired progression $\langle a,b\rangle_{m}$ is defined as

[TABLE]

The generalisation $j_{2}(n)$ of the Jacobsthal function with respect to pairedprogressions is called paired Jacobsthal function. It is defined to be the smallest naturalnumber $m$, such that every paired progression of length $m$, where the difference of its pair elements is even, contains at least one pair coprime to $n$ [4].

Definition 3**.**

For $n\in\mathbb{N}$, the paired Jacobsthal function $j_{2}(n)$ is defined as

[TABLE]

This function is also determined by its values for products of distinct primes [4], and the case of primorial numbers is most interesting. The function values at these points generate a separate function.

Definition 4**.**

For $n\in\mathbb{N}$, the primorial paired Jacobsthal function $h_{2}(n)$ is defined as

[TABLE]

It was proven that the conjectured upper bound $h_{2}(n)<p_{n}^{2}-p_{n},\ n\geq 3$ of this function represents a sufficient condition for the truth of Goldbach’s conjecture and of the prime pairs conjecture as well [4].

2 Computation and results

Several algorithmic concepts were successfully applied to the computation ofJacobsthal’s common function for primorials [3]. For the computation of the primorial paired Jacobsthal function, we extended and adapted these approaches for the case of paired progressions. A detailed analysis of how to compute $h_{2}(n)$ is provided in an ancillary file.

We computed the entire values of the function $h_{2}(n)$ for $n\leq 21$ or $p_{n}\leq 73$, respectively. The results for $3\leq n\leq 21$ fulfil the conjectured bound $h_{2}(n)<p_{n}^{2}-p_{n}$ [4]. The data are listed in table 1, including the maximum processed prime $p_{n}$, the values of $h_{2}(n)$, and the conjectured bound $p_{n}^{2}-p_{n}$.

In addition to this note, we provide several ancillary files. The file \dqfull_details.pdf\dq includes a detailed description of theory, algorithms, and further results. This paperwidely follows the structure of our former paper [3]. Furthermore, it outlines thecontent of all other anxillary data files representing exhaustive lists of the computational results.

Contact

marioziller@arcor.de

axelmorack@live.com