# 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.

## Key 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.