Infinite matrix of odd natural numbers. A bit about Sophie Germain prime numbers

TL;DR

This paper introduces an infinite matrix of natural numbers using a pairing function, providing new insights into prime distributions, including Sophie Germain primes, and offers a proof of the infinity of primes based on Bertrand's postulate.

Contribution

It presents a novel matrix construction of natural numbers that simplifies prime analysis and establishes the asymptotic density of Sophie Germain primes as 1/6.

Findings

The matrix separates odd and even numbers, aiding prime analysis.

Proves the infinity of primes using shell numbers and Bertrand's postulate.

Shows the asymptotic density of Sophie Germain primes is 1/6.

Abstract

In this paper we work with number sequences from the On-Line Encyclopedia of Integer Sequences (OEIS). Using the Pepis-Kalmar pairing function, we obtain an infinite matrix of natural numbers in which odd natural numbers are separated from even ones; such a matrix simplifies working with prime numbers. With point's shell numbers and shell lines we give another proof of the infinity of primes, based on Bertrand's postulate. We have shown that the asymptotic density of Mersenne numbers (OEIS A000225) is positive and equal to an infinitesimal value. It is also proven that Sophie Germain prime numbers (OEIS A005384) are located in the set of natural numbers with asymptotic density of 1/6.