Remarkable and Reversible Prime Number Patterns

TL;DR

This paper explores extended prime number patterns, including negative integers and prime powers, revealing new classifications, prime septets, and the density of generalized twin primes within these classes.

Contribution

It introduces novel classifications of prime patterns extending to negative integers and prime powers, and demonstrates the density of generalized twin primes.

Findings

Prime number septets at equal distances are identified.

Each class of generalized twin primes contains a positive fraction of all prime pairs.

Prime patterns are extended to negative integers and prime powers.

Abstract

Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each class of generalized twin primes of the classification contains a positive fraction of all prime pairs.