Reduced Collatz Dynamics is Periodical and the Period equals 2 to the power of the count of x/2

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preprint

posted on 2020-01-28, 03:36 authored by Wei RenWei Ren

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).

History

0000-0001-8590-1737

China University of Geosciences (Wuhan)