Patterns of Primes and Composites from Divisibility Network of Natural Numbers

TL;DR

This paper explores the divisibility network of natural numbers to uncover patterns in primes and composites, deriving analytical centrality measures and validating them through network analysis.

Contribution

It introduces a novel divisibility network framework for analyzing natural number patterns and provides analytical expressions for centrality measures related to primes and composites.

Findings

Derived analytical expressions for centrality measures using divisor functions

Validated measures with adjacency matrix methods

Revealed insights into prime and composite number patterns

Abstract

We present the pattern underlying some of the properties of natural numbers, using the framework of complex networks. The network used is a divisibility network in which each node has a fixed identity as one of the natural numbers and the connections among the nodes are made based on the divisibility pattern among the numbers. We derive analytical expressions for the centrality measures of this network in terms of the floor function and the divisor functions. We validate these measures with the help of standard methods which make use of the adjacency matrix of the network. Thus how the measures of the network relate to patterns in the behaviour of primes and composite numbers becomes apparent from our study.