Using Iterated Function Systems to Reveal Biases in the Distribution of Prime Numbers

TL;DR

This paper uses iterated function systems to uncover fractal patterns and biases in the distribution of prime numbers, revealing phenomena related to prime congruences and recent prime bias research.

Contribution

The paper introduces a novel application of IFS to visualize and analyze biases in prime distributions, connecting fractal patterns with prime number theory.

Findings

Fractal patterns indicating biases among primes with specific properties

Revealed 'repulsive' phenomena in prime distributions

Linked observed patterns to recent prime bias studies

Abstract

Iterated function systems (IFS) can be a surprisingly useful tool for studying structure in data. Here we present results stemming from a 2013 computational study by the author using IFS. The results include fractal patterns that reveal "repulsive" phenomena among primes in a wide range of classes, having specified arithmetic or congruence properties. Some of the phenomena shown in our computations relate to the recent, groundbreaking work of Lemke Oliver and Soundararajan on biases between consecutive primes. We do not have asymptotics to explain our results, but provide graphs, data, and detailed explanations of the phenomena.