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                                                      please read first  :       2.)    ►    The Ordered Distribution of Natural Numbers on the Square Root Spiral      

  extract from the study :     

Abstract :

Prime Numbers 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. To learn more about these amazing facts, see my detailed introduction to the Square Root Spiral :

“ The ordered distribution of natural numbers on 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.

With the kind permission of Mr Robert Sachs, I show some sections of his webside : www.numberspiral.com in this study. These sections contain interesting diagrams, which are related to my analysis results, especially in regards to the distribution of prime numbers.