A General Class of Collatz Sequence and Ruin Problem
In this paper we show the probabilistic convergence of the original Collatz (3n + 1) (or Hotpo) sequence to unity. A generalized form of the Collatz sequence (GCS) is proposed subsequently. Unlike Hotpo, an instance of a GCS can converge to integers other than unity. A GCS can be generated using the...
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Main Authors | , |
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Format | Journal Article |
Language | English |
Published |
09.03.2012
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Subjects | |
Online Access | Get full text |
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Summary: | In this paper we show the probabilistic convergence of the original Collatz
(3n + 1) (or Hotpo) sequence to unity. A generalized form of the Collatz
sequence (GCS) is proposed subsequently. Unlike Hotpo, an instance of a GCS can
converge to integers other than unity. A GCS can be generated using the concept
of an abstract machine performing arithmetic operations on different numerical
bases. Original Collatz sequence is then proved to be a special case of GCS on
base 2. The stopping time of GCS sequences is shown to possess remarkable
statistical behavior. We conjecture that the Collatz convergence elicits
existence of attractor points in digital chaos generated by arithmetic
operations on numbers. We also model Collatz convergence as a classical ruin
problem on the digits of a number in a base in which the abstract machine is
computing and establish its statistical behavior. Finally an average bound on
the stopping time of the sequence is established that grows linearly with the
number of digits. |
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DOI: | 10.48550/arxiv.1203.2229 |