The 3n+1 Problem: A Probabilistic Approach
The Poisson probability distribution is used to model the number of cycles
in the generalized Collatz problem.
First, interrelated cycles are defined and used as a criterion
in counting the cycles for a given q value.
Initially, archived data in the
mathematical literature (giving the known 3n + q cycles)
is analyzed. For
large samples, the Poisson probability distribution gives a poor fit of the
data (there are too many cycles for the large x values).
are defined and used as an additional criterion in counting cycles; this
improves the data fit substantially. Some theory and empirical results are
given in an attempt to explain the origin of this distribution of cycle
counts. Degrees of freedom in probability distributions involving the
difference between the number of odd and even elements in a cycle are
shown to be a partial explanation for the distribution of cycle counts.
(L, K) trees (generalized associated associated cycles) are defined and
used to account for the smallest difference between the number of odd
and even elements in the cycles for a given q
value. The article consists
entirely of analysis of empirical results; no proofs are given.
Full version: pdf,
(Concerned with sequence
Received February 20 2012;
revised version received May 22 2012.
Published in Journal of Integer Sequences, May 28 2012.
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