Circularity In The Collatz Sequence

             Bradley Berg                          December 30, 2022
                                         Updated:   January 14, 2023


                                ABSTRACT

       We consider the possibility that the Collatz sequence reaches the
       starting seed value and forms a loop.  To algebraically define when
       a sequence repeats we use:

            2^Even = 3^Odd + DL / Seed    where:
 
               Even is the number of even steps.
                Odd is the number of odd steps.
               DL is a sum that is derived from the sequence.

       The point where the sequence comes nearest to reaching the seed is
       when 3^Odd is close to 2^Even.  In binary close implies that 3^Odd
       has many lead one bits.  The carry from adding DL / Seed
       causes them to flip resulting in a power of two.  This constraint
       eliminates nearly all candidate loop lengths except when the number
       of Odd steps has many leading one bits in 3^Odd.

       A large seed is needed to produce a run with enough Odd steps so 
       that 3^Odd has even a modest number of leading one bits.  However,
       the width of the seed is also limited by the number of ones.  This
       limit is so low that no seed can produce a run that is big enough
       to form a loop.

1.0 Introduction

1.1 Expand the Sequence

1.2 Cycle Spotting

1.3 Leading Ones


1.4 Powers of Three

2.0 Loop Bounds In Binary

2.1 Size of DL

We can plug the maximum D value into the constraint to get the maximum Seed value for a given loop length.


The Seed is maximized when: KL = [log2( 3^L )] + 1 = width( 3^L )


2.2 Maximum Aligned Seed

2.3 Seed Limit Using Binary Arithmetic

Maximum Allowed Seed Width

2.4 Run Length

3.0 Conclusion