Band In One
A New Perspective on the Collatz Conjecture
The Collatz sequence has been intriguing and annoying mathematicians for nearly a hundred years. Beginning with
any positive integer n and using two elementary functions, 3n + 1 for odd numbers, and 2n for even numbers, you can create sequences of integers of variable length that invariably crash to a 4, 2, 1, 4…loop. This has been computationally verified by computer up to numbers in excess of 10^20. Anyone who has explored these sequences could testify to the elusiveness of consistent pattern in their unfolding, except of course for that final loop. Yet as persistent as the loop appears, it has not been demonstrated that it is inevitable for all positive integers.
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In approaching this problem, I have investigated reverse sequences. I have been able to construct Minimum Track Sequences (cMTS) that mirror in reverse segments of Collatz sequences. Most importantly, I am able to demonstrate that all Collatz sequences must run (in reverse) on one or more cMTS segments, a critical fact which generates additional parameters that condition the probabilities. Moreover, I can prove that there can be no other loop besides the 4, 2, 1, 4…loop. Finally, based on the “neighborhoods” of numbers associated with each term of a cMTS, I am able to define the Reverse Collatz Structure (RCS), a hierarchical structure with infinite and separate domains. However, while I can mount a strong probabilistic argument that all Collatz sequences must run in the RCS, (and thus are condemned to the 4-2-1-4 loop), I have not been able to provide a rigorous proof that that they are intrinsically a part of the RCS. But perhaps someone with a broader mathematical background can find a solution based on this new set of ideas.
These ideas are fully developed in my main paper, New Perspectives on the Collatz Sequence.pdf. You can download / read the file here.
In addtion to New Perspectives, I have created a number of resource files. Summaries of their salient information are included in New Perspectives. However, I have provided links to the files for detailed perusal.
Regular patterns are hard to come buy in Collatz sequences, but I found an elemental pattern in simple a number table that relates to high neighbors, a critical concept in the RCS. The file, Analysis of a Number Table Pattern, documents this pattern with mathematical proofs for its necessity.
In a Mathematica file, I explored patterns in cMTS. The samples included 50 K sequences in five separate ranges, form 10^6 to 10^30. The file, cMTS Profile.pdf, is a slightly edited version of the original Mathematica file.
In another Mathematica file, I explored patterns in Collatz sequence. In particular I focused on the statistical profile of the cMTS segments that make up a Collatz sequence. I sampled 60K numbers in ranges from 10^6 to 10^26. The file, Collatz Profile.pdf, is a slightly edited version of the original Mathematica file.
The slide show below provides a primer for the essential concepts I have created in exploring Collatz sequences, reverse sequences (cMTS), and the Reverse Collatz Structure (RCS).
