Version 1
: Received: 15 May 2022 / Approved: 16 May 2022 / Online: 16 May 2022 (11:04:46 CEST)
How to cite:
Pan, H. Special Spirals are Produced by the ROTASE Galactic Spiral Equations with the Sequential Prime Numbers. Preprints2022, 2022050207. https://doi.org/10.20944/preprints202205.0207.v1
Pan, H. Special Spirals are Produced by the ROTASE Galactic Spiral Equations with the Sequential Prime Numbers. Preprints 2022, 2022050207. https://doi.org/10.20944/preprints202205.0207.v1
Pan, H. Special Spirals are Produced by the ROTASE Galactic Spiral Equations with the Sequential Prime Numbers. Preprints2022, 2022050207. https://doi.org/10.20944/preprints202205.0207.v1
APA Style
Pan, H. (2022). Special Spirals are Produced by the ROTASE Galactic Spiral Equations with the Sequential Prime Numbers. Preprints. https://doi.org/10.20944/preprints202205.0207.v1
Chicago/Turabian Style
Pan, H. 2022 "Special Spirals are Produced by the ROTASE Galactic Spiral Equations with the Sequential Prime Numbers" Preprints. https://doi.org/10.20944/preprints202205.0207.v1
Abstract
In this paper, the sequential prime numbers are used as variables for the galactic spiral equations which were developed from the ROTASE model. Special spiral patterns are produced when prime numbers are treated with the unit of radian. The special spiral patterns produced with the first 1000 prime numbers have 20 spirals arranged in two groups. The two groups have perfect central symmetry with each other and are separated with two spiral gaps. The special spiral pattern produced with natural numbers from 1 to 7919 shows 6 spirals in the central area and 44 spirals in the outer area. The whole 7919 spiral points can be plotted with either 6-spiral pattern or 44-spiral pattern. For the spirals only produced by the prime numbers in the 6-spiral pattern plotting, the spiral 2 and spiral 3 each has only one spiral point produced by prime number 2 and 3, respectively, all other spiral points produced by other prime numbers are located on the spiral 1 and spiral 5. The special spiral pattern is well explained with careful analysis, it is concluded that all prime numbers greater than 3 must meet one of the equations: P1 = 1 + 6 * n (n > 0) P5 = 5 + 6 * n (n ≥ 0) In other words, every prime number greater than 3 is either a P1 prime number or a P5 prime number, no exception. Matching one of the equations is a necessary condition for a number to be a prime number, not a sufficient condition. Hope such sufficient condition can be found in the future. The number of P1 prime numbers roughly equal the number of P5 prime number in the first 2 billion prime numbers. The galactic spiral equations with golden angle can duplicate Vogel’s result for the simulation of sunflower seed head pattern, and a pinwheel pattern can be produced also with galactic spiral equations and 1 degree more than golden angle.
Keywords
prime numbers; galactic spiral equations; ROTASE model; phyllotaxis; sunflower seed head
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
