Circularity In The Collatz Sequence

             Bradley Berg                                  December 30, 2022
                                                   Updated:    July 29, 2023

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

2.2 Maximum Aligned Seed

2.3 Seed Limit Using Binary Arithmetic

2.4 Run Length

3.0 Conclusion


Appendix A: Short Run Lengths

Appendix B: Finding Near Misses

Appendix C: Cycles In The 3*n-1 Sequence

Appendix D: Cycles in 3n+C

Using the equality, here are some loops for C under 100:

Here are two longer loops with a lengths of 336 and 366: