Cycles and Divergent Trajectories for a Class of Permutation Sequences

John Simons

doi:doi:10.11648/j.ijtam.20220805.11

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Cycles and Divergent Trajectories for a Class of Permutation Sequences

John Simons

Published in International Journal of Theoretical and Applied Mathematics (Volume 8, Issue 5)

Received: 15 September 2022 Accepted: 13 October 2022 Published: 2 November 2022

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Abstract

Let f be a permutation from onto . Let x and consider a (finite or infinite) sequence s = (x, f(x), f²(x),•••). We call s a permutation sequence. Let D be the set of elements of s. If D is a finite set then the sequence s is a cycle, and if D is an infinite set the sequence s is a divergent trajectory. We derive theoretical and computational bounds for cycles and divergent trajectories for a defined class of permutations. This class contains generalizations of the original Collatz permutation f(2n) = 3n, f(4n + 1) = 3n + 1, f(4n + 3) = 3n + 2 and is based on a generalization of the covering system of congruences of Erdös. For the derivation of theoretical cycle bounds from transcendental number theory we adjust the cycle concept and the approach of Simons and de Weger for the 3x + 1 problem. For the derivation of computational cycle bounds we use continued fraction approximation methods. For a defined subclass the existence of divergent trajectories is proved.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 8, Issue 5)
DOI	10.11648/j.ijtam.20220805.11
Page(s)	85-95
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2022. Published by Science Publishing Group

Keywords

Permutation Sequences, Cyclic Trajectories, Divergent Trajectories, Collatz Conjecture, Transcendental Number Theory, Complete Coverage Sets

References

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[3]	T. Brox, Collatz cycles with few descents, Acta Arithmetica 92 (2000), pp. 181-88.
[4]	S. L. G. Choi, Covering the set of integers by congruence classes of distinct moduli, Math. Comp., 25 [1971], pp. 885-895.
[5]	R. E. Crandall, On the “3x + 1” problem, Math. Comp. 32 [1978], pp. 1281-1292.
[6]	J. L. Davidson, Some comments on an Iteration problem, Proceedings of the Sixty Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Manitoba, 1976 pp. 155-159, Congressus Numerantium XVII, Utilitas Math., Winnipeg, MN, 1977.
[7]	R. K. Guy, Unsolved Problems in Number Theory, second edition (1994) p. 219.
[8]	G. H. Hardy and E. M. Wright, “An Introduction to the Theory of Numbers”, 5th ed., Oxford University Press, 1981.
[9]	M. Hindry and J. H. Silverman, Diophantine Geometry, An Introduction, GTM 201, Springer Verlag, Berlin, 2000.
[10]	J. C. Lagarias, The 3x + 1 problem and its generalizations, Amer. Math. Monthly 85 (1985).
[11]	D. Levy, Injectivity and surjectivity of Collatz functions, Discrete Mathematics 285 [2004] pp. 191-199.
[12]	Michel Laurent, Maurice Mignotte and Yuri Nesterenko, “Formes linéaires en deux logarithmes et d´ eterminants d’interpolation”, J. Number Th. 55 [1995], pp. 285-321.
[13]	K. . Matthews, Generalized 3x + 1 mappings: Markov chains and ergodic theory, The Ultimate Challenge: The 3x + 1 Problem, (J. C. Lagarias, Ed.), pp. 79-104, Amer. Math. Society, Providence RI 2010.
[14]	S. Porubsky and J. Sch¨ onheim, The covering systems of Paul Erdös: Past, present and future, Paul Erd¨ os and his mathematics, Vol. I, Janos Bolyai and his mathematics, vol. I, Janos Bolyai Math. Soc., Budapest 2002, pp. 581- 627.
[15]	S. Porubsky, Results and problems on covering systems of residue classes, Mitteilungen aus dem Math. Sem. Giessen, Heft 150, Univ. Giessen 1981.
[16]	G. Rhin, “Approximants de Padé et mesures effectives d’irrationalité”, Progress in Mathematics 71 [1987], pp. 155-164.
[17]	R. P. Steiner, “A Theorem on the Syracuse Problem”, Proc. 7th Manitoba Conference on Numerical Mathematics 1977, Winnipeg 1978, pp. 553-559.
[18]	M. Tavares, “A remark about the density of the orbits of the Collatz permutation”, Discrete Mathematics 313 (2013), pp. 2139-2145.
[19]	J. L. Simons and B. M. M. de Weger, Theoretical and computationalboundsform-cyclesofthe3n+1problem. Acta arithmetica 117.1 [2005] pp. 51-70, see for latest results http://www.win.tue.nl/bdeweger/research.html.
[20]	G. Venturini, On a generalization of the 3x + 1 problem, Adv. Appl. Math. 19 [1997], pp. 295-305.

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APA Style

John Simons. (2022). Cycles and Divergent Trajectories for a Class of Permutation Sequences. International Journal of Theoretical and Applied Mathematics, 8(5), 85-95. https://doi.org/10.11648/j.ijtam.20220805.11

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ACS Style

John Simons. Cycles and Divergent Trajectories for a Class of Permutation Sequences. Int. J. Theor. Appl. Math. 2022, 8(5), 85-95. doi: 10.11648/j.ijtam.20220805.11

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John Simons. Cycles and Divergent Trajectories for a Class of Permutation Sequences. Int J Theor Appl Math. 2022;8(5):85-95. doi: 10.11648/j.ijtam.20220805.11

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@article{10.11648/j.ijtam.20220805.11,
  author = {John Simons},
  title = {Cycles and Divergent Trajectories for a Class of Permutation Sequences},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {8},
  number = {5},
  pages = {85-95},
  doi = {10.11648/j.ijtam.20220805.11},
  url = {https://doi.org/10.11648/j.ijtam.20220805.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220805.11},
  abstract = {Let f be a permutation from  onto . Let x   and consider a (finite or infinite) sequence s = (x, f(x), f2(x),•••). We call s a permutation sequence. Let D be the set of elements of s. If D is a finite set then the sequence s is a cycle, and if D is an infinite set the sequence s is a divergent trajectory. We derive theoretical and computational bounds for cycles and divergent trajectories for a defined class of permutations. This class contains generalizations of the original Collatz permutation f(2n) = 3n, f(4n + 1) = 3n + 1, f(4n + 3) = 3n + 2 and is based on a generalization of the covering system of congruences of Erdös. For the derivation of theoretical cycle bounds from transcendental number theory we adjust the cycle concept and the approach of Simons and de Weger for the 3x + 1 problem. For the derivation of computational cycle bounds we use continued fraction approximation methods. For a defined subclass the existence of divergent trajectories is proved.},
 year = {2022}
}

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TY  - JOUR
T1  - Cycles and Divergent Trajectories for a Class of Permutation Sequences
AU  - John Simons
Y1  - 2022/11/02
PY  - 2022
N1  - https://doi.org/10.11648/j.ijtam.20220805.11
DO  - 10.11648/j.ijtam.20220805.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 85
EP  - 95
PB  - Science Publishing Group
SN  - 2575-5080
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AB  - Let f be a permutation from  onto . Let x   and consider a (finite or infinite) sequence s = (x, f(x), f2(x),•••). We call s a permutation sequence. Let D be the set of elements of s. If D is a finite set then the sequence s is a cycle, and if D is an infinite set the sequence s is a divergent trajectory. We derive theoretical and computational bounds for cycles and divergent trajectories for a defined class of permutations. This class contains generalizations of the original Collatz permutation f(2n) = 3n, f(4n + 1) = 3n + 1, f(4n + 3) = 3n + 2 and is based on a generalization of the covering system of congruences of Erdös. For the derivation of theoretical cycle bounds from transcendental number theory we adjust the cycle concept and the approach of Simons and de Weger for the 3x + 1 problem. For the derivation of computational cycle bounds we use continued fraction approximation methods. For a defined subclass the existence of divergent trajectories is proved.
VL  - 8
IS  - 5
ER  -

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Author Information

John Simons

Department of Systems Engineering, University of Groningen, Groningen, The Netherlands

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John Simons. (2022). Cycles and Divergent Trajectories for a Class of Permutation Sequences. International Journal of Theoretical and Applied Mathematics, 8(5), 85-95. https://doi.org/10.11648/j.ijtam.20220805.11

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ACS Style

John Simons. Cycles and Divergent Trajectories for a Class of Permutation Sequences. Int. J. Theor. Appl. Math. 2022, 8(5), 85-95. doi: 10.11648/j.ijtam.20220805.11

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AMA Style

John Simons. Cycles and Divergent Trajectories for a Class of Permutation Sequences. Int J Theor Appl Math. 2022;8(5):85-95. doi: 10.11648/j.ijtam.20220805.11

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@article{10.11648/j.ijtam.20220805.11,
  author = {John Simons},
  title = {Cycles and Divergent Trajectories for a Class of Permutation Sequences},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {8},
  number = {5},
  pages = {85-95},
  doi = {10.11648/j.ijtam.20220805.11},
  url = {https://doi.org/10.11648/j.ijtam.20220805.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220805.11},
  abstract = {Let f be a permutation from  onto . Let x   and consider a (finite or infinite) sequence s = (x, f(x), f2(x),•••). We call s a permutation sequence. Let D be the set of elements of s. If D is a finite set then the sequence s is a cycle, and if D is an infinite set the sequence s is a divergent trajectory. We derive theoretical and computational bounds for cycles and divergent trajectories for a defined class of permutations. This class contains generalizations of the original Collatz permutation f(2n) = 3n, f(4n + 1) = 3n + 1, f(4n + 3) = 3n + 2 and is based on a generalization of the covering system of congruences of Erdös. For the derivation of theoretical cycle bounds from transcendental number theory we adjust the cycle concept and the approach of Simons and de Weger for the 3x + 1 problem. For the derivation of computational cycle bounds we use continued fraction approximation methods. For a defined subclass the existence of divergent trajectories is proved.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Cycles and Divergent Trajectories for a Class of Permutation Sequences
AU  - John Simons
Y1  - 2022/11/02
PY  - 2022
N1  - https://doi.org/10.11648/j.ijtam.20220805.11
DO  - 10.11648/j.ijtam.20220805.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 85
EP  - 95
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20220805.11
AB  - Let f be a permutation from  onto . Let x   and consider a (finite or infinite) sequence s = (x, f(x), f2(x),•••). We call s a permutation sequence. Let D be the set of elements of s. If D is a finite set then the sequence s is a cycle, and if D is an infinite set the sequence s is a divergent trajectory. We derive theoretical and computational bounds for cycles and divergent trajectories for a defined class of permutations. This class contains generalizations of the original Collatz permutation f(2n) = 3n, f(4n + 1) = 3n + 1, f(4n + 3) = 3n + 2 and is based on a generalization of the covering system of congruences of Erdös. For the derivation of theoretical cycle bounds from transcendental number theory we adjust the cycle concept and the approach of Simons and de Weger for the 3x + 1 problem. For the derivation of computational cycle bounds we use continued fraction approximation methods. For a defined subclass the existence of divergent trajectories is proved.
VL  - 8
IS  - 5
ER  -

Copy | Download