An artificial life view of the collatz problem

  • Authors:

  • Hiroki Sayama

  • Affiliations:

  • Departments of Bioengineering & Systems Science and Industrial Engineering, Binghamton University, State University of New York, Binghamton, New York

  • Venue:
  • Artificial Life
  • Year:
  • 2011

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Abstract

This letter presents a new, artificial-life-based view of the Collatz problem, a well-known mathematical problem about the behavior of a series of positive integers generated by a simple arithmetical rule. The Collatz conjecture asserts that this series always falls into a 4 → 2 → 1 cycle regardless of its initial values. No formal proof has been given yet. In this letter, the behavior of the series is considered an ecological process of artificial organisms (Is in bit strings). The Collatz conjecture is then reinterpreted as the competition between population growth and extinction. This new interpretation has made it possible to analytically calculate the growth and extinction speeds of bit strings. The results indicate that the extinction is alway s faster than the growth, providing an ecological explanation for the conjecture. Future research directions are also suggested.