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Register a new userPatterns of Primes and Composites from Divisibility Network of Natural Numbers
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We present the pattern underlying some of the properties of natural numbers, using the framework of complex networks. The network used is a divisibility network in which each node has a fixed identity as one of the natural numbers and the connections among the nodes are made based on the divisibility pattern among the numbers. We derive analytical expressions for the centrality measures of this network in terms of the floor function and the divisor functions. We validate these measures with the help of standard methods which make use of the adjacency matrix of the network. Thus how the measures of the network relate to patterns in the behaviour of primes and composite numbers becomes apparent from our study.
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We study the undirected divisibility graph in which the vertex set is a finite subset of consecutive natural numbers up to N.We derive analytical expressions for measures of the graph like degree, clustering, geodesic distance and centrality in terms of the floor functions and the divisor functions. We discuss how these measures depend on the vertex labels and the size of graph N. We also present the specific case of prime vertices separately as corollaries. We could explain the patterns in the local measures for a finite size graph as well as the trends in global measures as the size of the graph increases.
Natural numbers can be divided in two non-overlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as building blocks remains elusive. Here, we propose a new approach to decoding the architecture of natural numbers based on complex networks and stochastic processes theory. We introduce a parameter-free non-Markovian dynamical model that naturally generates random primes and their relation with composite numbers with remarkable accuracy. Our model satisfies the prime number theorem as an emerging property and a refined version of Cramers conjecture about the statistics of gaps between consecutive primes that seems closer to reality than the original Cramers version. Regarding composites, the model helps us to derive the prime factors counting function, giving the probability of distinct prime factors for any integer. Probabilistic models like ours can help to get deeper insights about primes and the complex architecture of natural numbers.
We prove some theorems which give sufficient conditions for the existence of prime numbers among the terms of a sequence which has pairwise relatively prime terms.
Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the Square Root Spiral (also named as the Spiral of Theodorus or Wurzel Spirale or Einstein Spiral). Prime Numbers also clearly accumulate on such spiral graphs. And the square numbers 4, 9, 16, 25, 36,... form a highly three-symmetrical system of three spiral graphs, which divides the square-root-spiral into three equal areas. A mathematical analysis shows that these spiral graphs are defined by quadratic polynomials. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. Fibonacci Numbers divide the Square Root Spiral into areas and angle sectors with constant proportions. These proportions are linked to the golden mean (or golden section), which behaves as a self-avoiding-walk-constant in the lattice-like structure of the square root spiral.
The natural numbers divisible by the Prime Factors 2, 3, 5, 11, 13 and 17 lie on defined spiral graphs, which run through the Square Root Spiral. A mathematical analysis shows, that these spiral graphs are defined by specific quadratic polynomials. Basically all natural number which are divisible by the same prime factor lie on such spiral graphs. And these spiral graphs can be assigned to a certain number of Spiral Graph Systems, which have a defined spatial orientation to each other. This document represents a supplementation to my detailed introduction study to the Square Root Spiral, and it contains the missing diagrams and analyses, showing the distribution of the natural numbers divisible by 2, 3, 5, 11, 13 and 17 on the Square Root Spiral. My introduction study to the Square Root Spiral can be found in the arxiv-archive. The title of this study : The ordered distribution of the natural numbers on the Square Root Spiral.
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