Multiplicative largeness of $\textit{de Polignac numbers}$

TL;DR

This paper proves that the set of de Polignac numbers, related to differences of consecutive primes, is both additively and multiplicatively syndetic, using combinatorial and ultrafilter techniques.

Contribution

It establishes that de Polignac numbers form a multiplicative syndetic set, extending previous additive results with new multiplicative insights.

Findings

De Polignac numbers form an additively syndetic set.

De Polignac numbers form a multiplicatively syndetic set.

Uses ultrafilter and combinatorial methods in the proof.

Abstract

A number $m$ is said to be a $\textit{de Polignac number}$, if infinitely many pairs of consecutive primes exist, such that $m$ can be written as the difference of those consecutive prime numbers. Recently in [ W. D. Banks: Consecutive primes and IP sets, arXiv:2403.10637.], using arguments from the Ramsey theory, W. D. Banks proved that the collection of $\textit{de Polignac number}$ is an $IP^\star$ set (Though his original statement is relatively weaker, an iterative application of pigeonhole principle/ theory of ultrafilters shows that this statement is sufficient to conclude the set is $IP^\star$). As a consequence, we have this collection as an additively syndetic set. In this article, we show that this collection is also a multiplicative syndetic set. In our proof, we use combinatorial arguments and the tools from the algebra of the Stone-\v{C}ech compactification of discrete semigroups (for details see [N. Hindman, and D. Strauss: Algebra in the Stone-\v{C}ech Compactification: Theory and Applications, second edition, de Gruyter, Berlin,2012.]).