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Collatz Dynamics is Partitioned by Residue Class Regularly
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Version 1
: Received: 5 September 2022 / Approved: 13 September 2022 / Online: 13 September 2022 (03:21:20 CEST)
A peer-reviewed article of this Preprint also exists.
Ren, W. Collatz Dynamics Is Partitioned by Residue Class Regularly. Research in Mathematics 2023, 10, doi:10.1080/27684830.2023.2269657. Ren, W. Collatz Dynamics Is Partitioned by Residue Class Regularly. Research in Mathematics 2023, 10, doi:10.1080/27684830.2023.2269657.
Abstract
We propose Reduced Collatz Conjecture that is equivalent to Collatz Conjecture, which states that every positive integer can return to an integer less than it, instead of 1. Reduced Collatz Conjecture should be easier because some properties are presented in reduced dynamics, rather than in original dynamics (e.g., ratio and period). Reduced dynamics is a computation sequence from starting integer to the first integer less than it, and original dynamics is a computation sequence from starting integer to 1. Reduced dynamics is a component of original dynamics. We denote dynamics of x as a sequence of either computations in terms of ``I'' that represents (3*x+1)/2 and ``O'' that represents x/2. Here 3*x+1 and x/2 are combined together, because 3*x+1 is always even and followed by x/2. We formally prove that all positive integers are partitioned into two halves and either presents ``I'' or ``O'' in next ongoing computation. More specifically, (1) if any positive integer x that is i module $2^t$ (i is an odd integer) is given, then the first t computations (each one is either ``I'' or ``O'' corresponding to whether current integer is odd or even) will be identical with that of i. (2) If current integer after t computations (in terms of ``I'' or ``O'') is less than x, then reduced dynamics of x is available. Otherwise, the residue class of x (namely, i module $2^t$) can be partitioned into two halves (namely, i module $2^{t+1}$ and $i+2^t$ module $2^{t+1}$), and either half presents ``I'' or ``O'' in intermediately forthcoming (t+1)-th computation.
Keywords
Collatz conjecture; 3x+1 problem; Dynamics System; Residue Class
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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