Multi-scale representation of integer sets: application to prime numbers

TL;DR

This paper introduces a multi-scale hierarchical analysis method for studying arithmetic properties of integer sets, exemplified on prime numbers, revealing scale-invariant characteristics and simplifying large integer analysis.

Contribution

The paper presents a novel multi-scale framework that encodes integer properties into small integer sequences, enabling analysis of prime number distributions across different scales.

Findings

Histograms at various scales share a common invariant shape.

Encoding primes into small integers simplifies large integer analysis.

The method reveals scale-invariant properties of prime distributions.

Abstract

We propose a multi-scale analysis method for studying arithmetic properties of integer sets, such as primality. Our approach organizes information through a hierarchy of nested sequences, where each level enables a hierarchical expression of the studied property by examining patterns at varying levels of granularity. To illustrate the method, we apply it to prime numbers. While this does not claim any new breakthroughs on this classical problem, the approach allows for analysis of the studied property across large integer sequences and reveals characteristics observable at different scales. By limiting ourselves to the case of prime numbers, we build sequences with values in {0, ..., 255}, which have the advantage of simplifying the reading, at different scales, of the encoded property. We free ourselves from the numerous digits of large integers by replacing them with small integers between 0 and 255. We have also highlighted, at different scales, histograms composed of at most 256 values. We have observed that for a sufficiently large interval, they all share a same invariant shape, which can be viewed as a characteristic of prime numbers. Each value in the histogram represents the count of a subset of prime numbers. We have proposed an estimation for each value in the histogram and at all scales. We hope that the proposed framework will be useful for investigating arithmetic properties.