A Reduced Forward Collatz Algorithm: How Binary Strings Change Their Length Under 3x+1

TL;DR

This paper introduces an algorithm that analyzes how binary representations of odd numbers change under the reduced Collatz map, providing insights into number growth, shrinkage, and potential counterexamples.

Contribution

The paper presents a new algorithm for tracking binary digit changes in the reduced Collatz map, simplifying the analysis of odd number transformations.

Findings

Algorithm determines growth or shrinkage of binary length for odd numbers

Provides a method to identify potential smallest counterexamples to Collatz conjecture

Simplifies analysis of binary digit patterns in Collatz iterations

Abstract

We developed an algorithm that easily goes from one odd number to the next odd number in binary representation for the reduced forward Collatz map (Syracuse function). The algorithm indicates when an odd number can grow or shrink to the next odd number based on the pattern of binary digits. The algorithm is also used to provide a simpler method for determining the change in binary string length for the reduced map than one found in the literature. Accordingly, an inspection of the binary digits for an odd number can determine the number of binary digits of the subsequent odd number. We also show some simple results for what the smallest number could be for a counterexample to the Collatz conjecture.