Algorithm speed-ups of 3x+1 convergence verification sieves

Research Article
Open access

Algorithm speed-ups of 3x+1 convergence verification sieves

Kevin Ge 1*
  • 1 W. T. Woodson High School    
  • *corresponding author kevinge0086@gmail.com
TNS Vol.5
ISSN (Print): 2753-8826
ISSN (Online): 2753-8818
ISBN (Print): 978-1-915371-53-9
ISBN (Online): 978-1-915371-54-6

Abstract

The 3x+1 Conjecture is a notorious open elementary number theory problem for its deceitful triviality. Several algorithms described in the past for the numerical verification for the convergence, one of which implemented an improvement of sieves. However, most papers focused on theoretical aspects of the sieve, and empirical results were rarely discussed. This paper presents data from implementing sieves on a relatively high-level language, as well as some helpful unmentioned rudimentary algebraic facts about sieves. Several algorithms for convergence verification of the 3x+1 Problem had been developed since the 1970s. This paper tests an algorithm presented in a 1999 paper and explores the runtime growth to input size.

Keywords:

3x + 1 problem, collatz conjecture, sieves, number theory

Ge,K. (2023). Algorithm speed-ups of 3x+1 convergence verification sieves. Theoretical and Natural Science,5,960-965.
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References

[1]. Leavens, G. T., & Vermeulen, M. (1992). 3x+ 1 search programs. Computers & Mathematics with Applications, 24(11), 79-99.

[2]. Lagarias, J. C. (2003). The 3x+ 1 problem: An annotated bibliography (1963–1999). The ultimate challenge: the 3x, 1, 267-341.

[3]. Lagarias, J. C. (2006). The 3x+ 1 problem: An annotated bibliography, II (2000-2009). arXiv preprint math/0608208.

[4]. Eliahou, S. (1993). The 3x+ 1 problem: new lower bounds on nontrivial cycle lengths. Discrete mathematics, 118(1-3), 45-56.

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

[6]. Oliveira e Silva, T. (2010). Empirical verification of the 3x+1 and related conjectures. The ultimate challenge: The 3x+1 problem, 189-207.

[7]. Bařina, D. (2021). Convergence verification of the Collatz problem. The Journal of Supercomputing, 77(3), 2681-2688. doi: 10. 1007 / s11227 - 020 - 03368-x.

[8]. Bařina, D. (2021). url: https://pcbarina.fit.vutbr.cz/.

[9]. Oliveira e Silva, T. (1999). Maximum excursion and stopping time record-holders for the 3x+ 1 problem: computational results. Mathematics of Computation, 68(225), 371-384.

[10]. Roosendaal, E., On the 3x + 1 problem, url: http://www.ericr.nl/wondrous/index.html.

Cite this article

Ge,K. (2023). Algorithm speed-ups of 3x+1 convergence verification sieves. Theoretical and Natural Science,5,960-965.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

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About volume

Volume title: Proceedings of the 2nd International Conference on Computing Innovation and Applied Physics (CONF-CIAP 2023)

ISBN：978-1-915371-53-9(Print) / 978-1-915371-54-6(Online)
Editor：Marwan Omar, Roman Bauer
Conference website: https://www.confciap.org/
Conference date: 25 March 2023
Series: Theoretical and Natural Science
Volume number: Vol.5
ISSN：2753-8818(Print) / 2753-8826(Online)

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References

[1]. Leavens, G. T., & Vermeulen, M. (1992). 3x+ 1 search programs. Computers & Mathematics with Applications, 24(11), 79-99.

[2]. Lagarias, J. C. (2003). The 3x+ 1 problem: An annotated bibliography (1963–1999). The ultimate challenge: the 3x, 1, 267-341.

[3]. Lagarias, J. C. (2006). The 3x+ 1 problem: An annotated bibliography, II (2000-2009). arXiv preprint math/0608208.

[4]. Eliahou, S. (1993). The 3x+ 1 problem: new lower bounds on nontrivial cycle lengths. Discrete mathematics, 118(1-3), 45-56.

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

[6]. Oliveira e Silva, T. (2010). Empirical verification of the 3x+1 and related conjectures. The ultimate challenge: The 3x+1 problem, 189-207.

[7]. Bařina, D. (2021). Convergence verification of the Collatz problem. The Journal of Supercomputing, 77(3), 2681-2688. doi: 10. 1007 / s11227 - 020 - 03368-x.

[8]. Bařina, D. (2021). url: https://pcbarina.fit.vutbr.cz/.

[9]. Oliveira e Silva, T. (1999). Maximum excursion and stopping time record-holders for the 3x+ 1 problem: computational results. Mathematics of Computation, 68(225), 371-384.

[10]. Roosendaal, E., On the 3x + 1 problem, url: http://www.ericr.nl/wondrous/index.html.

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