System(3X±1)/4 Framework for the Collatz (3x+1) Problem

Abstract

In order to gain deeper insight into the Collatz (3x+1) Problem, we design a framework based on System(3X±1)/4; defined similar to the Collatz (3x+1) system, with slightly modified operations.  On similar lines, we define the CTUHSK(3x+1)/4 system, and show that to be exactly the same as the Collatz (3x+1) system.  The convergence of System(3X±1)/4 to its trivial-cycle {(1⇐(2)⇐4)} is established using the persistent local characteristic property of ‘odd-to-odd monotonicity’ in its defining function.  We also present a meticulously designed reversible (invertible) transformation (reduction); between System(3X±1)/4 and the CTUHSK(3x+1)/4 system; based on pruning-&-joining operations on the system arborescence; which does not affect the binary(inward)arborescence structure as well as the convergence characteristics of either of these systems.  This establishes the convergence of the CTUHSK(3x+1)/4 system to its trivial-cycle {(1⇐2⇐4)}.

This paper also presents Halemane-Conjecture associated with the Collatz sequence, that the upper bound on the number of odd (3x+1) operations required to reach its trivial-cycle {(1⇐2⇐4)} starting from any given positive integer and moving along the Collatz sequence, is the given number itself; the triad {(31⇐41⇐27)} being an exceptional limiting case.