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The Distribution of Prime Numbers on the Square Root Spiral

By Hahn, Harry K.

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Book Id: WPLBN0100301946
Format Type: PDF (eBook)
File Size: undefined
Reproduction Date: 6/30/2007

Title: The Distribution of Prime Numbers on the Square Root Spiral  
Author: Hahn, Harry K.
Language: English
Subject: Non Fiction, Science, prime number distribution , square root spiral
Collections: Mathematics, Authors Community
Publication Date:
Publisher: Self-published
Member Page: Harry Hahn


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Hahn, H. K. (2007). The Distribution of Prime Numbers on the Square Root Spiral. Retrieved from

The Square Root Spiral ( or “Spiral of Theodorus” or “Einstein Spiral” ) is a very interesting geometrical structure in which the square roots of all natural numbers have a clear defined orientation to each other. This enables the attentive viewer to find many spatial interdependencies between natural numbers, by applying simple graphical analysis techniques. Therefore the Square Root Spiral should be an important research object for professionals, who work in the field of number theory ! The most amazing property of the square root spiral is surely the fact, that the distance between two successive winds of the Square Root Spiral quickly strives for the well known geometrical constant Pi !! Mathematical proof that this statement is correct is shown in Chapter 1 “The correlation with Pi“ in the mathematical section of my detailed introduction to the Square Root Spiral : --> see previous study ! : “The ordered distribution of the natural numbers on the Square Root Spiral”

Prime Numbers clearly accumulate on defined spiral graphs,which run through the Square Root Spiral. These spiral graphs can be assigned to different spiral-systems, in which all spiral-graphs have the same direction of rotation and the same -second difference- between the numbers, which lie on these spiral-graphs. A mathematical analysis shows, that these spiral graphs are caused exclusively by quadratic polynomials. For example the well known Euler Polynomial x2+x+41 appears on the Square Root Spiral in the form of three spiral-graphs, which are defined by three different quadratic polynomials. All natural numbers,divisible by a certain prime factor, also lie on defined spiral graphs on the Square Root Spiral (or Spiral of Theodorus, or Wurzelspirale). And the Square Numbers 4, 9, 16, 25, 36 even form a highly three-symmetrical system of three spiral graphs, which divides the square root spiral into three equal areas. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. With the help of the Number-Spiral, described by Mr. Robert Sachs, a comparison can be drawn between the Square Root Spiral and the Ulam Spiral. The shown sections of his study of the number spiral contain diagrams, which are related to my analysis results, especially in regards to the distribution of prime numbers.

Another striking property of the Square Root Spiral is the fact, that the square roots of all square numbers ( 4, 9, 16, 25, 36… ) lie on 3 highly symmetrical spiral graphs which divide the square root spiral into 3 equal areas. ( see FIG.1 : graphs Q1, Q2 and Q3 drawn in green color ).

Table of Contents
44 pages, 26 figures, 7 tables


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