Monotonicity and Convergence in the Collatz Conjecture: A New Perspective

Resumen

The Collatz conjecture declares that every positive integer will eventually reach 1 when subjected to a simple iterative process: if the number is even, it is divided by 2, and if it is odd, it is multiplied by 3 and then increased by 1. Despite the straightforward nature of these rules, a general proof of the conjecture remains elusive. For the above, this study introduces an alternative interpretation of the conjecture. This approach involves multiplying an odd integer N₁ by 3 and subsequently adding the largest power-of-2 factor within N₁. Repeated iterations of this alternative process show that any initial odd integer N₁ will eventually convert into a power of 2, leading the sequence towards convergence. The behavior of the sequence was studied by representing the resulting integers as a power of 2 multiplied by an odd component. Using this representation under the modified rules, we developed a structured proof framework that demonstrates the consistent reduction of the odd component’s relative value after each iteration, the accelerated increase of the power-of-2 factor’s relative value, and the absence of any divergent cycles or alternative behaviors. This analysis provides insights into the mechanics of convergence in the Collatz sequence and proposes a new perspective for understanding the conjecture’s underlying dynamics.