**Lawrence Somer and Michal Křížek**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 30, 2024, Number 1, Pages 47–66

DOI: 10.7546/nntdm.2024.30.1.47-66

**Full paper (PDF, 257 Kb)**

## Details

### Authors and affiliations

Lawrence Somer

*Department of Mathematics, Catholic University of America
Washington, D.C. 20064, United States*

Michal Křížek

*Institute of Mathematics, Czech Academy of Sciences
Žitná 25, CZ — 115 67 Prague 1, Czech Republic*

### Abstract

Let be an odd prime and let and be two Lucas sequences whose discriminants have the same nonzero quadratic character modulo and whose periods modulo are equal. We prove that there is then an integer such that for all , the frequency with which appears in a full period of is the same frequency as appears in . Here satisfies the recursion relation with initial terms and . Similar results are obtained for the companion Lucas sequences and . This paper extends analogous statements for Lucas sequences of the form given in a previous article. We further generalize our results by showing for a certain class of primes that if , , and and are Lucas sequences with the same period modulo , then there exists an integer such that for all residues , the frequency with which appears in is the same frequency as appears in .

### Keywords

- Lucas sequences
- Discriminant
- Primes
- Second-order recurrence

### 2020 Mathematics Subject Classification

- 11B39
- 11A07
- 11A41

### References

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### Manuscript history

- Received: 8 September 2023
- Revised: 21 February 2024
- Accepted: 23 February 2024
- Online First: 27 February 2024

### Copyright information

Ⓒ 2024 by the Authors.

This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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## Cite this paper

Somer, L., & Křížek, M. (2024). Second-order linear recurrences with identically distributed residues modulo *p ^{e}*.

*Notes on Number Theory and Discrete Mathematics*, 30(1), 47-66, DOI: 10.7546/nntdm.2024.30.1.47-66.