Divisibility in paired progressions, Goldbach's conjecture, and the infinitude of prime pairs

TL;DR

This paper explores progressions in integer pairs, introduces a generalized Jacobsthal function, and shows that a specific upper bound could imply the truth of Goldbach's conjecture, twin prime infinitude, and related prime pair conjectures.

Contribution

It proposes a generalized Jacobsthal function and links its conjectured upper bound to key unsolved problems in prime number theory.

Findings

Conjecture of a specific upper bound for the generalized Jacobsthal function.

Proof that this bound would imply Goldbach's conjecture.

Proof that this bound would imply the infinitude of twin primes.

Abstract

We investigate progressions in the set of pairs of integers $\mathbb{Z}^2$ and define a generalisation of the Jacobsthal function. For this function, we conjecture a specific upper bound and prove that this bound would be a sufficient condition for the truth of the Goldbach conjecture, the infinitude of prime twins, and more general of prime pairs with a fixed even difference.