Structures in P based on Properties of Semigroup and Arithmetical Sequence H = (+-3*2; 1)

TL;DR

This paper explores the structure of prime numbers within a specific arithmetical sequence H, revealing its properties as a semigroup, its relation to prime twins, and providing insights into the Goldbach conjecture.

Contribution

It introduces a new framework based on the sequence H for analyzing prime structures, including a revised definition of P and geometric interpretations of prime relations.

Findings

H is the smallest superset of P excluding 3 and 2

H forms a semigroup with unique properties

The set of prime twins in H is infinite

Abstract

This paper presents results on structures in P based on tools developed from subjects of elementary number theory. Key findings are: The arithmetical sequence H = (+-3*2; 1) is in Z the smallest superset of P \ {3, 2}. H is a semigroup. A revised definition of P. Unique Gestalt of p in Z. The prime number lattice packing H exp n. The geometrical locus in HxH of the family of solutions of: - the set of prime twins, - the set of PRACHAR prime twins, - in H exp 2, H exp 3 family of solutions of the GOLDBACH conjunction. Partition of H in p exp 2-intervals. Prime numbers in p exp 2-intervals. Infinity of the set of prime twins. Verification of the GOLDBACH conjunction.