Complex Network Approach to Number Theory

TL;DR

This paper introduces a novel complex network-based model for number theory that simulates prime distributions using bipartite graphs and differential equations, potentially aiding in understanding prime behavior and solving open problems.

Contribution

It presents a new algorithm for creating bipartite graphs of integers that closely mimic real prime distributions, linking complex network theory with number theory.

Findings

The model accurately reproduces prime number distribution patterns.

Differential equations describe the simulated prime distribution.

Potential to prove open number theory questions using the model.

Abstract

In this short paper, following the most recent advances in complex network theory, a new approach to number theory with potential applications to other fields is proposed. The model by Garcia-Perez, Serrano and Boguna, introduces an algorithm which allows to create a bipartite graph of integers (with primes and composites) statistically very close to the real one. Since the algorithm is defined a priori, we can have a description of the simulated prime number distribution in terms of a known differential equation, which in general can be treated more easily. The so determined properties of the simulated distribution can give useful hints about the behavior of the real prime number distribution. In principle it could be also possible to demonstrate open questions in number theory, proven the total equivalence of the simulated and real distributions.