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
Abstract
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).
