A Definitive Proof of the Collatz Conjecture using the Halemane(3X±1)/4System

Abstract

Halemane(3X±1)/4System is defined, similar to the Collatz (3x+1) System, with slightly modified operations.  On similar lines, we define the CTUHSK(3x+1)/4System, and show that to be exactly the same as the Collatz (3x+1) System.  The convergence of Halemane(3X±1)/4System to its trivial-cycle {(1⇐(2)⇐4)} is established.  A reversible (invertible) transformation - based on a judiciously designed pruning-&-joining operations on the system arborescence - between Halemane(3X±1)/4System and CTUHSK(3x+1)/4System is shown not to affect the binary(inward)arborescence structure as well as the convergence characteristics of either of the systems.  This establishes the convergence of the CTUHSK(3x+1)/4System and therefore provides a definitive proof of the convergence of the Collatz-Thwaites-Ulam-Hasse-Syracuse-Kakutani (CTUHSK) Sequence; asserting the Collatz Conjecture.

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 the 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.