Using Binary and Bezout Identity to Prove Collatz Conjecture

We propose a novel framework utilizing a full binary tree structure to systematically represent the set of natural numbers, which we classify into three subsets: pure odd numbers, pure even numbers, and mixed numbers. Within this framework, we employ a binary string representation for natural numbers and develop a comprehensive composite methodology that integrate both odd- and even-number functions. Our investigation centers on the itera- tive dynamics of the Collatz function and its reduced variant, which effectively serves as a pruning mechanism for the full binary tree, enabling rigorous exam- ination of the Collatz conjecture’s validity. To establish a robust foundation for this conjecture, we ingeniously incorporate binary strings into an algebraic formulation that fundamentally captures the intrinsic properties of the Col- latz sequence. Through this analytical framework, we demonstrate that the sequence generated by infinite iterations of the Collatz function constitutes an eventually periodic sequence, thereby providing a discuss of this long-standing mathematical conjecture that has remained unresolved for 87 years.