Binary Representation of Natural Numbers and Collatz Conjecture

We propose a novel framework utilizing a full binary tree structure to systemati- 1 cally represent the set of natural numbers, which we classify into three subsets: pure odd 2 numbers, pure even numbers, and mixed numbers. Within this framework, we employ a 3 binary string representation for natural numbers and develop a comprehensive composite 4 methodology that integrate both odd- and even-number functions. Our investigation 5 centers on the iterative dynamics of the Collatz function and its reduced variant, which 6 effectively serves as a pruning mechanism for the full binary tree, enabling rigorous ex- 7 amination of the Collatz conjecture’s validity. To establish a robust foundation for this 8 conjecture, we ingeniously incorporate binary strings into an algebraic formulation that 9 fundamentally captures the intrinsic properties of the Collatz sequence. Through this 10 analytical framework, we demonstrate that the sequence generated by infinite iterations 11 of the Collatz function constitutes an eventually periodic sequence, thereby providing 12 a rigorous validation of this long-standing mathematical conjecture that has remained 13 unresolved for 87 years.