Three theorems on twin primes

TL;DR

This paper explores the relationship between a specific integer sequence and twin primes, providing new theorems, a sufficient condition for infinitely many twin primes, and proposing conjectures and an algorithm for constructing twin primes.

Contribution

It introduces three theorems connecting sequence A166944 with twin primes, offers a new sufficient condition for their infinitude, and proposes a simple conjectural algorithm for generating twin primes.

Findings

Proved three theorems relating sequence A166944 to twin primes.

Provided a sufficient condition for the infinity of twin primes.

Suggested a conjectural algorithm for constructing twin primes.

Abstract

For earlier considered our sequence A166944 in [4] we prove three statements of its connection with twin primes. We also give a sufficient condition for the infinity of twin primes and pose several new conjectures; among them we propose a very simple conjectural algorithm of constructing a pair $(p,\enskip p+2)$ of twin primes over arbitrary given integer $m\geq4$ such that $p+2\geq m.$