Some computational results on a conjecture of de Polignac about numbers of the form p + 2k

Abstract

Let $U$ be the set of positive odd numbers that can not be written in the form $p+2^k$. Recently, by analyzing possible prime divisors of b, Chen proved $b\ge 11184810$ and $\omega \left(b\right)\ge 7$ if an arithmetic progression $a\left(\mathrm{mod}b\right)$ is in $U$, with $\omega \left(b\right)=7$ if and only if $b=11184810$, where $\omega \left(n\right)$ is the number of distinct prime divisors of n. In this paper, we take a computational approach to prove $b\ge 11184810$ and provide all possible values of a if $a\left(\mathrm{mod}11184810\right)$ is in $U$. Moreover, we explicitly construct nontrivial arithmetic progressions $a\left(\mathrm{mod}b\right)$ in $U$ with $\omega \left(b\right)=8$, 9, 10, or 11, and provide potential nontrivial arithmetic progressions $a\left(\mathrm{mod}b\right)$ in $U$ such that $\omega \left(b\right)=s$ for any fixed $s\ge 12$. Furthermore, we improve the upper bound estimate of numbers of the form $p+2^k$ by Habsieger and Roblot in 2006 to 0.490341088858244 by enhancing their algorithm and employing GPU computation.