Improved verification limit for the convergence of the Collatz conjecture

Abstract This article presents our project, which aims to verify the Collatz conjecture computationally. As a main point of the article, we introduce a new result that pushes the limit for which the conjecture is verified up to $$2^{71}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mn>71</mml:mn> </mml:msup> </mml:math> . We present our baseline algorithm and then several sub-algorithms that enhance acceleration. The total acceleration from the first algorithm we used on the CPU to our best algorithm on the GPU is $$1\,335\times$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mspace/> <mml:mn>335</mml:mn> <mml:mo>×</mml:mo> </mml:mrow> </mml:math> . We further distribute individual tasks to thousands of parallel workers running on several European supercomputers. Besides the convergence verification, our program also checks for path records during the convergence test. We found four new path records.