Reduced Collatz Dynamics is Periodical and the Period equals 2 to the power of the count of x/2
Abstract
We propose Reduced Collatz Conjecture that is equivalent to Collatz
Conjecture but is easier to explore, because reduced dynamics is
more primitive than original dynamics and presents better structures
(e.g., period and ratio). Reduced dynamics (that are occurred
computation sequence from a starting integer to the first integer
less than the starting integer) is the component of original
dynamics (from a starting integer to 1). Reduced dynamics of x is
represented by a sequence of computation that is either (3*x+1)/2 or
x/2, because 3*x+1 is always even and followed by x/2. We prove that
reduced dynamics is periodical and its period equals 2 to the power
of the count of x/2. More specifically, if there exists reduced
dynamics of x, then there exists reduced dynamics of x+P, where P
equals $2^L$ and L is the total count of x/2 computations in
reduced
dynamics of x (equivalently, L is the length of the sequence).
