**Mustafa Ismail, Salah Eddine Rihane and M. Anwar**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 29, 2023, Number 3, Pages 462–473

DOI: 10.7546/nntdm.2023.29.3.462-473

**Full paper (PDF, 202 Kb)**

## Details

### Authors and affiliations

Mustafa Ismail

*Department of Mathematics, Faculty of Science,
University of Ain Shams, Egypt*

Salah Eddine Rihane

*Department of Mathematics, Institute of Science and Technology,
University Center of Mila, Algeria*

M. Anwar

*Department of Mathematics, Faculty of Science,
University of Ain Shams, Egypt*

### Abstract

Let be the Narayana sequence defined by the recurrence for all with intital values and . In this paper, we fully characterize the -adic valuation of and and then we find all positive integer solutions to the Brocard–Ramanujan equation where is a Narayana number.

### Keywords

- Narayana sequence
- Factorials
*p*-adic valuation

### 2020 Mathematics Subject Classification

- 11B39
- 11D72

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### Manuscript history

- Received: 20 July 2022
- Revised: 25 April 2023
- Accepted: 27 June 2023
- Online First: 6 July 2023

### Copyright information

Ⓒ 2023 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).

## Related papers

- Marques, D. (2012). Fibonacci numbers at most one away from the product of factorials.
*Notes on Number Theory and Discrete Mathematics*, 18(3), 13–19.

## Cite this paper

Ismail, M., Rihane, S. E., Anwar, M. (2023). Narayana sequence and the Brocard–Ramanujan equation. *Notes on Number Theory and Discrete Mathematics*, 29(3), 462-473, DOI: 10.7546/nntdm.2023.29.3.462-473.