A note on the Polignac numbers

TL;DR

This paper investigates the distribution of Polignac numbers within certain structured sets derived from admissible sets, establishing that these sets always contain at least one Polignac number under specified conditions.

Contribution

It proves that for large enough admissible sets, scaled differences always include at least one Polignac number, extending understanding of their distribution.

Findings

Sets derived from admissible sets contain Polignac numbers for large parameters.

The result holds for admissible sets with size at least 3.5 million.

Polignac numbers are guaranteed in scaled difference sets under the given conditions.

Abstract

Suppose that $k\geq 3.5\times 10^6$ and $\hH=\{h_1,\ldots,h_{k_0}\}$ is admissible. Then for any $m\geq 1$, the set $$ \{m(h_j-h_i):\,h_i<h_j\} $$ contains at least one Polignac number.