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Prime numbers and random walks in a square grid.

Alberto Fraile1, Osame Kinouchi2, Prashant Dwivedi1

  • 1Department of Control Engineering, Faculty of Electrical Engineering, Czech Technical University in Prague, Karlovo náměstí 13, 121 35, Czech Republic.

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This study introduces a prime walk algorithm, using prime number last digits to create deterministic grid walks. The resulting patterns reveal surprising regularities in two and three dimensions.

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Area of Science:

  • Number Theory
  • Computational Mathematics
  • Algorithmic Analysis

Background:

  • Computer simulations are increasingly vital for exploring prime number properties.
  • The distribution and patterns of prime numbers remain a key area of mathematical inquiry.
  • Understanding prime number sequences can lead to novel algorithmic structures.

Purpose of the Study:

  • To introduce a novel algorithm for a deterministic walk on a grid, termed a prime walk.
  • To investigate the structural properties and patterns generated by this prime walk algorithm.
  • To analyze the regularities present in the two- and three-dimensional structures created by the walk.

Main Methods:

  • Definition of a deterministic walk algorithm on a two-dimensional grid.
  • Construction of the walk sequence based on the last digits of prime numbers.
  • Analysis of the resulting geometric patterns in two and three dimensions.

Main Results:

  • The prime walk algorithm generates structures with notable properties and regularities.
  • Despite using a seemingly random sequence (last digits of primes), the generated patterns exhibit order.
  • Analysis reveals consistent characteristics in both 2D and 3D representations of the prime walk.

Conclusions:

  • The prime walk algorithm offers a new method for visualizing and analyzing prime number sequences.
  • The observed regularities suggest underlying structures not apparent in the prime number sequence itself.
  • This approach provides a computational tool for exploring the intricate nature of prime numbers.