How Prime Factors Form Fractals

TL;DR

This work introduces a division-free sieve that simultaneously identifies primes and outputs full prime factorizations, revealing that the generated sequences are the $p$-adic valuations $v_p(n)$ and are fractal. By formalizing fractal criteria as predictable self-containment and aperiodicity, the paper demonstrates that all $v_p\langle\rangle$ sequences possess fractal structure, and illustrates deep geometric connections by showing $v_2\langle\rangle$ encodes the Lévy Dragon and the odd-part sequence $o_n\langle\rangle$ encodes the Heighway Dragon (mod 4). The results unify number-theoretic factorization with fractal geometry, offering a new perspective on how prime exponents distribute across integers and how these distributions manifest as classic fractal curves. The work further explores the implications for other primes, raising questions about angle choices and the universality of fractal behavior across $v_p(n)$ sequences. Overall, the paper provides a novel framework linking sieve-based factorization, $p$-adic valuations, and fractal geometry, with potential pedagogical and mathematical implications.

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.

Figures (20)

• Figure 1: The number line as a "tree."
• Figure 2: The number line, with 90 degree turns representing $v_{2}\langle\rangle$.
• Figure 3: The number line (without numbers) with 90-degree turns representing $v_{2}(n)$.
• Figure 4: A curve with 90-degree turns representing https://oeis.org/A346070.
• Figure 5: A curve with 90-degree turns representing $v_{2}\langle\rangle$.

Theorems & Definitions (25)

• Proposition 1
• proof
• Proposition 2
• proof
• Lemma 3
• proof
• Proposition 4
• proof
• Lemma 5
• proof