How Prime Factors Form Fractals

TL;DR

This paper introduces a novel sieve method for generating primes and prime factorizations without division, revealing that these sequences are fractal and relate to well-known fractals like the Levy and Heighway Dragons.

Contribution

The paper presents a new prime sieve that produces fractal sequences and connects prime factorization to geometric fractals, offering a novel perspective on number theory and fractal geometry.

Findings

Sequences are p-adic valuations of n

Generated sequences form fractals like Levy Dragon

Connection established between odd part of n and Heighway Dragon

Abstract

We explore a new sieve that generates both primes and prime factorizations, without resorting to division. We demonstrate that the integer sequences generated by the sieve are the p-adic valuations of n, and that each is a fractal sequence. We then show that these sequences produce geometrical fractals like the Levy Dragon. We end by showing the connection between the odd part of n integer sequence and the Heighway Dragon.