10.36227/techrxiv.11664567.v1
Wei Ren
Wei
Ren
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
TechRxiv
2020
Collatz conjecture
2020-01-23 05:32:23
Preprint
https://www.techrxiv.org/articles/preprint/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_ln2/11664567
<div>We propose Reduced Collatz conjecture and prove that it is</div><div>equivalent to Collatz conjecture but more primitive due to reduced</div><div>dynamics. We study reduced dynamics (that consists of occurred</div><div>computations from any starting integer to the first integer less</div><div>than it), because it is the component of original dynamics (from any</div><div>starting integer to 1). Reduced dynamics is denoted as a sequence of</div><div>``I'' that represents (3*x+1)/2 and ``O'' that represents x/2. Here</div><div>3*x+1 and x/2 are combined together because 3*x+1 is always even and</div><div>thus followed by x/2. We discover and prove two key properties on</div><div>reduced dynamics: (1) Reduced dynamics is invertible. That is, given</div><div>a reduced dynamics, a residue class that presents such reduced</div><div>dynamics, can be computed directly by our derived formula. (2)</div><div>Reduced dynamics can be constructed algorithmically, instead of by</div><div>computing concrete 3*x+1 and x/2 step by step. We discover the</div><div>sufficient and necessary condition that guarantees a sequence</div><div>consisting of ``I'' and ``O'' to be a reduced dynamics. Counting</div><div>from the beginning of a sequence, if and only if the count of x/2</div><div>over the count of 3*x+1 is larger than ln3/ln2, reduced dynamics</div><div>will be obtained (i.e., current integer will be less than starting</div><div>integer).</div>