An Explicit Ramsey Bound for de Polignac Numbers

Abstract

A number $ m $ is called a \textbf{de Polignac number}($ \mathbf{POL} $ in short) if it can be expressed as the difference of infinitely many pairs of consecutive prime numbers. For any given $r\in \mathbb N,$ a set $A$ is said to be $Δ_r^\star$ if for all sets $S$ with $|S|=r$ such that $A\cap \{s-t:s>t\in S\}\neq \emptyset.$ In this article we prove that the set $\mathbf{POL}$ is $Δ_r^\star$ with the specific computable value of $r,$ where $r=exp(\mathcal{O}(50)).$ Then we prove that there exists a set $E$ with $|E|\leq 2^{50\cdot \prod_{i=1}^{49}p_i} $ (where $(p_i)_i$ is the enumeration of primes) such that $\mathbb{N}=\bigcup_{t\in E} t^{-1}\mathbf{POL}.$