A Reduced Collatz Dynamics Maps to a Residue Class, and Its Count of x/2 over the 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 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).