Collatz Dynamics is Partitioned by Residue Class Regularly

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.

31 Dec 2023