An improved algorithm for checking the Collatz conjecture for all n < 2^N

TL;DR

This paper introduces a more efficient algorithm for verifying the Collatz conjecture up to 2^N and discusses its extension to negative numbers, significantly improving verification speed for larger ranges.

Contribution

The paper presents a novel algorithm that reduces verification time for the Collatz conjecture across exponentially increasing ranges of n.

Findings

Verification time roughly doubles when increasing N by 1

Algorithm effectively extends to negative numbers

Improved computational efficiency for large N

Abstract

We describe a new algorithm for verifying the Collatz conjecture for all n < 2^N for some fixed N. The algorithm takes less than twice as long to verify convergence for all n < 2^{N+1} as it does to verify convergence for all n < 2^N. We also discuss verification of the analogue of the Collatz conjecture for negative numbers.