Collatz Dynamics is Partitioned by Residue Class Regularly

Abstract

We propose Reduced Collatz Conjecture that is equivalent to Collatz Conjecture, which states that every positive integer can return to an integer less than it, instead of 1. Reduced Collatz Conjecture is easier to explore because certain structures must be presented in reduced dynamics, rather than in original dynamics (as original dynamics is a mixture of original dynamics). Reduced dynamics is a computation sequence from starting integer to the first integer less than it, in terms of ``I'' that represents (3*x+1)/2 and ``O'' that represents x/2. We formally prove that all positive integers are partitioned into two halves and either presents ``I'' or ``O'' in next ongoing computation. More specifically, (1) if any positive integer x that is i module $2^t$ (i is an odd integer) is given, then the first t computations (each one is either ``I'' or ``O'' corresponding to whether current integer is odd or even) will be identical with that of i. (2) If current integer after t computations (in terms of ``I'' or ``O'') is less than x, then reduced dynamics of x is available. Otherwise, the residue class of x (namely, i module $2^t$) can be partitioned into two halves (namely, i module $2^{t+1}$ and $i+2^t$ module $2^{t+1}$), and either half presents ``I'' or ``O'' in intermediately forthcoming (t+1)-th computation. This discovery will be helpful to the final proof of Collatz conjecture - if the union of residue classes who present reduced dynamics that become larger with the growth of residue module, equals all positive integers asymptotically, then Reduced Collatz Conjecture (or equivalently, Collatz Conjecture) will be true.