Theory of Computing Systems https://doi.org/10.1007/s00224-020-09986-5 Multiplication Algorithm Based on Collatz Function David Barina 1 © Springer Science+Business Media, LLC, part of Springer Nature 2020 Abstract This article presents a new multiplication algorithm based on the Collatz function. Assuming the validity of the Collatz conjecture, the time complexity of multiplying two n-digit numbers is O(kn), where the k is the number of odd steps in the Collatz trajectory of the first multiplicand. Most likely, the algorithm is only of theoretical interest. Keywords Multiplication algorithm · Division algorithm · Computer arithmetic · Collatz conjecture 1 Introduction One of the most famous problems in number theory that remains unsolved is the Collatz conjecture, which asserts that, for arbitrary positive integer x , a sequence defined by repeatedly applying the function C(x) =  3x + 1 if x is odd, or x/2 if x is even (1) will always converge to the cycle passing through the number 1. The terms of such sequence typically rise and fall repeatedly, oscillate wildly, and grow at a dizzying pace. The conjecture has never been proven. There are however experimental evi- dence [1] and heuristic arguments [2–4] that support it. There is also an extensive literature, [5, 6], on this question. Quoting Chamberland [7], some authors work with the Collatz function, Z + odd → Z + odd , defined by F(x) = (3x + 1)/ 2 ctz(3x+1) , (2)  David Barina ibarina@fit.vutbr.cz 1 Brno University of Technology, Faculty of Information Technology, Centre of Excellence IT4Innovations, Brno, Czech Republic