GPU-accelerated Exhaustive Verification of the Collatz Conjecture

Abstract

The main contribution of this paper is to present an implementation that performs the exhaustive search to verify the Collatz conjecture using a GPU. Consider the following operation on an arbitrary positive number: if the number is even, divide it by two, and if the number is odd, triple it and add one. The Collatz conjecture asserts that, starting from any positive number m, repeated iteration of the operations eventually produces the value 1. We have implemented it on NVIDIA GeForce GTX TITAN X and evaluated the performance. The experimental results show that, our GPU implementation can verify 1.31 × 10 12 64-bit numbers per second. While the sequential CPU implementation on Intel Core i7-4790 can verify 5.25×10 9 64-bit numbers per second. Thus, our implementation on the GPU attains a speed-up factor of 249 over the sequential CPU implementation. Additionally, we accelerated the computation of counting the number of the above operations until a number reaches 1, called delay that is one of the mathematical interests for the Collatz conjecture by the GPU. Using a similar idea, we achieved a speed-up factor of 73.

Computational Verification of the Collatz Problem

“…Also, in 2017, Honda et al [10] claimed they can check 2 40.25 numbers per second on the GPU. Their program is, however, only able to verify 64-bit numbers.…”

Inflation Propensity of Collatz Orbits: A New Proof-of-Work for Blockchain Applications

“…Current computational capabilities have allowed confirming the conjecture for very large numbers. For example, [8] introduced a GPU-based method to verify the Collatz algorithm. The authors could verify 1.31 × 10 12 64-bit numbers per second.…”