On Primorial Numbers

TL;DR

This paper explores properties of primorial numbers, introduces related concepts like primorial sets and intervals, and investigates their connections to prime tuples and the Goldbach conjecture.

Contribution

It defines new primorial-related structures and establishes relationships with prime numbers and conjectures, offering fresh perspectives in number theory.

Findings

Relationships between prime numbers and primorial totative numbers

Connections between admissible prime tuples and primorial tuples

Formulation of four new Goldbach conjectures

Abstract

Prime numbers have attracted the attention of mathematiciansand enthusiasts for millenniums due to their simple definition and remarkable properties. In this paper, we study primorial numbers (the product of the first prime numbers) to define primorial sets, primorial intervals, primorial tables, and primorial totative numbers. We establish relationships between prime numbers and primorial totative numbers and between admissible k-tuples of prime numbers and admissible k-tuples of primorial totative. Finally, we study the Goldbach conjecture and derive four Goldbach conjectures using primordial intervals, twin, cousin, and sexy prime numbers.