Skip to main content

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 \(O(2^N)\) 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.

This is a preview of subscription content, access via your institution.





  4. Enhancement proposed by Eric Roosendaal.



  1. Hercher C (2018) Über die Länge nicht-trivialer Collatz-Zyklen. Die Wurzel 6 and 7

  2. Lagarias JC (2003) The \(3x+1\) problem: an annotated bibliography (1963–1999) (sorted by author). arXiv:math/0309224

  3. Lagarias JC (2006) The \(3x+1\) problem: an annotated bibliography, II (2000–2009). arXiv:math/0608208

  4. Conway JH (1972) Unpredictable iterations. In: Proceedings of the 1972 Number Theory Conference, pp 49–52

  5. Mol LD (2008) Tag systems and Collatz-like functions. Theor Comput Sci 390(1):92–101.

    MathSciNet  Article  MATH  Google Scholar 

  6. Lagarias JC (1985) The 3x + 1 problem and its generalizations. Am Math Mon 92(1):3–23.

    MathSciNet  Article  MATH  Google Scholar 

  7. Honda T, Ito Y, Nakano K (2017) GPU-accelerated exhaustive verification of the Collatz conjecture. Int J Netw Comput 7(1):69–85

    Google Scholar 

  8. Oliveira e Silva T (2010) Empirical verification of the 3x+1 and related conjectures. In: Lagarias JC (ed) The ultimate challenge: The 3x+1 problem. American Mathematical Society, Providence, pp 189–207

    MATH  Google Scholar 

  9. Leavens GT, Vermeulen M (1992) 3x+1 search programs. Comput Math Appl 24(11):79–99.

    MathSciNet  Article  MATH  Google Scholar 

  10. Dunn R (1973) On Ulam’s problem. Tech. rep., University of Colorado at Boulder

  11. Oliveira e Silva T (1999) Maximum excursion and stopping time record-holders for the \(3x+1\) problem: computational results. Math Comput 68(225):371–384.

    MathSciNet  Article  MATH  Google Scholar 

  12. Lagarias JC, Weiss A (1992) The \(3x + 1\) problem: two stochastic models. Ann Appl Probab 2(1):229–261.

    MathSciNet  Article  MATH  Google Scholar 

Download references


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.”

Author information

Authors and Affiliations


Corresponding author

Correspondence to David Barina.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Barina, D. Convergence verification of the Collatz problem. J Supercomput 77, 2681–2688 (2021).

Download citation

  • Published:

  • Issue Date:

  • DOI:


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