On prime numbers of the form $2^n \pm k$

TL;DR

This paper explores the relationship between integers $k$ for which infinitely many primes satisfy $p+k$ being a power of 2 and the limit points of certain rational sets connected to polynomial sequences, advancing understanding of prime distributions.

Contribution

It establishes a novel connection between prime numbers of specific forms and the limit points of rational sets derived from polynomial sequences, providing new insights into prime number patterns.

Findings

Identifies a relationship between the set $\\mathcal{K}$ and limit points of rational sets.

Shows how polynomial sequences relate to primes of the form $2^n \\pm k$.

Provides a framework for analyzing primes linked to powers of two.

Abstract

Consider the set $\mathcal{K}$ of integers $k$ for which there are infinitely many primes $p$ such that $p+k$ is a power of $2$. The aim of this paper is to show a relationship between $\mathcal{K}$ and the limits points of some set rational numbers related to a sequence of polynomials $C_n(q)$ introduced by Kassel and Reutenauer [KasselReutenauer].