2024/06/04 by Goswami, Sayan
#05D10 #11E25 #11T30 #37A45 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2406.02243
A number m is said to be a 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 de Polignac number is an IP^⋆ 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^⋆). 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-Čech compactification of discrete semigroups (for details see [N. Hindman, and D. Strauss: Algebra in the Stone-Čech Compactification: Theory and Applications, second edition, de Gruyter, Berlin,2012.]).