Cycles of higher-order Collatz sequences

John L. Simons
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 1, Pages 48—63
DOI: 10.7546/nntdm.2022.28.1.48-63
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Authors and affiliations

John L. Simons
University of Groningen
PO Box 800, 9700 AV Groningen, The Netherlands


Consider a sequence of numbers x_n \in \mathbb{Z_+} defined by x_{n+1}= \frac{x_n}{2} if x_n is even, and x_{n+1}= \frac{x_n+2x_{n-1}+q}{2} if x_n is odd. A 1-cycle is a periodic sequence with one transition from odd to even numbers. We prove theoretical and computational results for the existence of 1-cycles, and discuss a generalization to more complex cycles.


  • Collatz problem
  • Higher order difference equation
  • Linear form in logarithms

2020 Mathematics Subject Classification

  • 11B83
  • 11J86


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Manuscript history

  • Received: 10 June 2021
  • Revised: 18 January 2022
  • Accepted: 2 February 2022
  • Online First: 9 February 2022

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Cite this paper

Simons, J. L. (2022). Cycles of higher-order Collatz sequences. Notes on Number Theory and Discrete Mathematics, 28(1), 48-63, DOI: 10.7546/nntdm.2022.28.1.48-63.

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