Efficient Computation of Collatz Sequence Stopping Times: A Novel Algorithmic Approach

TL;DR

This paper introduces a new algorithm that significantly improves the efficiency of computing Collatz sequence stopping times, enabling faster analysis of large numbers and advancing computational approaches to the conjecture.

Contribution

The paper presents a novel algorithm that reduces computational complexity in calculating Collatz stopping times, outperforming existing methods without relying on memoization or parallelization.

Findings

Achieves approximately 28% faster computation than previous methods

Handles extremely large numbers efficiently

Demonstrates scalability and robustness of the algorithm

Abstract

The Collatz conjecture, which posits that any positive integer will eventually reach 1 through a specific iterative process, is a classic unsolved problem in mathematics. This research focuses on designing an efficient algorithm to compute the stopping time of numbers in the Collatz sequence, achieving significant computational improvements. By leveraging structural patterns in the Collatz tree, the proposed algorithm minimizes redundant operations and optimizes computational steps. Unlike prior methods, it efficiently handles extremely large numbers without requiring advanced techniques such as memoization or parallelization. Experimental evaluations confirm computational efficiency improvements of approximately 28% over state-of-the-art methods. These findings underscore the algorithm's scalability and robustness, providing a foundation for future large-scale verification of the conjecture and potential applications in computational mathematics.