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

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.]).