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Download file# A Reduced Collatz Dynamics Maps to a Residue Class, and its Count of x/2 over Count of 3*x+1 is larger than ln3/ln2

We propose Reduced Collatz conjecture and prove that it is

equivalent to Collatz conjecture but more primitive due to reduced

dynamics. We study reduced dynamics (that consists of occurred

computations from any starting integer to the first integer less

than it), because it is the component of original dynamics (from any

starting integer to 1). Reduced dynamics is denoted as a sequence 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

thus followed by x/2. We discover and prove two key properties on

reduced dynamics: (1) Reduced dynamics is invertible. That is, given

a reduced dynamics, a residue class that presents such reduced

dynamics, can be computed directly by our derived formula. (2)

Reduced dynamics can be constructed algorithmically, instead of by

computing concrete 3*x+1 and x/2 step by step. We discover the

sufficient and necessary condition that guarantees a sequence

consisting of ``I'' and ``O'' to be a reduced dynamics. Counting

from the beginning of a sequence, if and only if the count of x/2

over the count of 3*x+1 is larger than ln3/ln2, reduced dynamics

will be obtained (i.e., current integer will be less than starting

integer).