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Divisibility patterns of natural numbers on a complex network.

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This study introduces a complex network model to analyze number divisibility. The research reveals scale-free network properties and a novel "stretching similarity" pattern in number sequences.

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Area of Science:

  • Number Theory
  • Network Science
  • Statistical Inference

Background:

  • Divisibility properties of natural numbers are a fundamental area of number theory.
  • Traditional methods for studying number patterns include various analytical and computational tools.
  • Complex network frameworks offer novel perspectives for analyzing mathematical structures.

Purpose of the Study:

  • To investigate the divisibility of natural numbers through the lens of a growing complex network.
  • To identify statistical properties and novel patterns within this network representation.
  • To analyze how network characteristics evolve with increasing network size.

Main Methods:

  • Construction of a growing complex network based on number divisibility.
  • Application of statistical inference tools to analyze network properties.
  • Investigation of local clustering patterns and global network metrics.
  • Analytical estimation and numerical validation of network behavior.

Main Results:

  • The complex network exhibits scale-free properties but with a non-stationary degree distribution.
  • A new local clustering pattern, termed "stretching similarity," was identified.
  • Network characteristics such as average degree, global clustering, and assortativity coefficient demonstrate smooth variation with network size.
  • Asymptotic behaviors of global clustering and average degree were analytically estimated and numerically confirmed.

Conclusions:

  • A complex network model provides a powerful framework for exploring number divisibility.
  • The identified network properties and patterns offer new insights into the structure of natural numbers.
  • The study validates the utility of network science and statistical inference in number theory research.