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Convergence verification of the Collatz problem

Convergence verification of the Collatz problem This article presents a new algorithmic approach for computational convergence verification of the Collatz problem. The main contribution of the paper is the replacement of huge precomputed tables containing O(2N)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(2^N)$$\end{document} entries with small lookup tables comprising just O(N) elements. Our single-threaded CPU implementation can verify 4.2×109\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$4.2 \times 10^9$$\end{document} 128-bit numbers per second on Intel Xeon Gold 5218 CPU computer, and our parallel OpenCL implementation reaches the speed of 2.2×1011\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2.2 \times 10^{11}$$\end{document} 128-bit numbers per second on NVIDIA GeForce RTX 2080. Besides the convergence verification, our program also checks for path records during the convergence test. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Journal of Supercomputing Springer Journals

Convergence verification of the Collatz problem

The Journal of Supercomputing , Volume 77 (3) – Jul 1, 2020

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References (13)

Publisher
Springer Journals
Copyright
Copyright © Springer Science+Business Media, LLC, part of Springer Nature 2020
ISSN
0920-8542
eISSN
1573-0484
DOI
10.1007/s11227-020-03368-x
Publisher site
See Article on Publisher Site

Abstract

This article presents a new algorithmic approach for computational convergence verification of the Collatz problem. The main contribution of the paper is the replacement of huge precomputed tables containing O(2N)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(2^N)$$\end{document} entries with small lookup tables comprising just O(N) elements. Our single-threaded CPU implementation can verify 4.2×109\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$4.2 \times 10^9$$\end{document} 128-bit numbers per second on Intel Xeon Gold 5218 CPU computer, and our parallel OpenCL implementation reaches the speed of 2.2×1011\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2.2 \times 10^{11}$$\end{document} 128-bit numbers per second on NVIDIA GeForce RTX 2080. Besides the convergence verification, our program also checks for path records during the convergence test.

Journal

The Journal of SupercomputingSpringer Journals

Published: Jul 1, 2020

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