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

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.