Recursive Approach for Proving Collatz Conjecture

Abstract

In this study, the authors investigate the Collatz conjecture using a frequency-based iterative approach, demonstrating that all natural numbers ultimately converge to a reduced value with an increasing frequency rate, eventually leading to a cyclic loop. Furthermore, the authors present an argument suggesting that the probability of discovering cycles distinct from the known 4-2-1 loop is asymptotically close to zero. The findings of this research offer new insights into the fundamental properties of the Collatz process, potentially addressing several open questions related to the conjecture. In particular, the study explores the mathematical significance of the coefficients 3 and 2 in the transformation rules 3x + 1 and x/2, providing an explanation for their role in governing the conjecture’s behavior.