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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## Details

### Authors and affiliations

**John L. Simons**

*University of Groningen
PO Box 800, 9700 AV Groningen, The Netherlands*

### Abstract

Consider a sequence of numbers defined by if is even, and if is odd. A -cycle is a periodic sequence with one transition from odd to even numbers. We prove theoretical and computational results for the existence of -cycles, and discuss a generalization to more complex cycles.

### Keywords

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

### 2020 Mathematics Subject Classification

- 11B83
- 11J86

### References

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### Manuscript history

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

## Related papers

## 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.