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
should be easier because some properties are presented in reduced
dynamics, rather than in original dynamics (e.g., ratio and period).
Reduced dynamics is a computation sequence from starting integer to
the first integer less than it, and original dynamics is a
computation sequence from starting integer to 1. Reduced dynamics is
a component of original dynamics. We denote dynamics of x as a
sequence of either computations in terms 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 followed by 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.
