The Origin of Natural Numbers and the 6k±1 Dual Tracks
Year: 2026
Type: preprint
Abstract: Starting from the intrinsic structure of natural numbers, this paper constructs a structural evolutionary path originating from "the source of numbers" to prime number orbits. Beginning with 1, the starting point of natural numbers, this study proposes that parity dichotomy is the most fundamental innate property of natural numbers. The constant emergence of 0.5 in triangular numbers and dimensional reduction transformation serves as the dichotomy critical point within the natural number system. On this basis, the 6k\pm1 dual-track structure is systematically illustrated: all primes greater than 3 only lie on the two orbits 6k-1 and 6k+1. The prime-composite ratio of the two orbits exhibits narrow-range oscillation, which is defined in this paper as the "dual-track breathing phenomenon". The model and observational data presented can offer a novel geometric observation pathway for the research on prime distribution. Rigorous analytical proof remains a subject for further follow-up investigation. Keywords: Origin of natural numbers; 6k\pm1 dual tracks; dichotomy critical point; prime-composite ratio breathing; structural evolutionary path
Author: Han, Jun
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214493481