Entropy Makes the Collatz Sequence Go Down

             Bradley Berg                          November 21, 2018
                                                Edited June 18, 2024


       We look at the Collatz Sequence from an information theory 
       perspective to lay out its underlying computational mechanics.  
       The mechanisms are similar to those used in pseudo random number 
       generators and one way hashes.

       An individual run is divided into three phases.  In the first
       phase the influence of the seed runs its course after information
       contained in the initial seed is lost.  Values are randomized in
       the second phase.  They follow the statistical model where the
       average gain is just over 0.866; causing them to decline.  The
       third phase begins once a value goes below the seed; providing
       that the series is not circular.  At this point we know the run
       will terminate at one.

       An equivalent restatement of the Collatz sequence steps through 
       alternating chains of even and odd values.  This variation
       constitutes a pseudo random number generator.  The operations
       used to scramble values are unbiased which results in an even
       distibution of ones and zeros.  The entropy of this mechanism is
       high so that in the second phase values are fairly randomized.

1.0 Introduction

1.1 Even and Odd Chains

1.2 Combining Even And Odd Chains

2.0 Sequence Entropy

2.1 Losing Information

2.2 Randomization Phase

2.3 Reduction To One

2.4 Observed Entropy

3.0 Conclusion