Convergence verification of the Collatz problem

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 \(O(2^N)\) entries with small lookup tables comprising just O(N) elements. Our single-threaded CPU implementation can verify \(4.2 \times 10^9\) 128-bit numbers per second on Intel Xeon Gold 5218 CPU computer, and our parallel OpenCL implementation reaches the speed of \(2.2 \times 10^{11}\) 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.

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Notes

  1. 1.

    http://www.ericr.nl/wondrous/.

  2. 2.

    https://boinc.thesonntags.com/collatz/.

  3. 3.

    http://sweet.ua.pt/tos/3x+1.html.

  4. 4.

    Enhancement proposed by Eric Roosendaal.

  5. 5.

    https://github.com/xbarin02/collatz/.

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Acknowledgements

Computational resources were supplied by the project “e-Infrastruktura CZ” (e-INFRA LM2018140) provided within the program Projects of Large Research, Development and Innovations Infrastructures. This work was supported by The Ministry of Education, Youth and Sports from the Large Infrastructures for Research, Experimental Development and Innovations project “IT4Innovations National Supercomputing Center – LM2015070.”

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Correspondence to David Barina.

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Barina, D. Convergence verification of the Collatz problem. J Supercomput 77, 2681–2688 (2021). https://doi.org/10.1007/s11227-020-03368-x

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Keywords

  • Collatz conjecture
  • Software optimization
  • Parallel computing
  • Number theory