Bakir Farhi
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 4, Pages 820–826
DOI: 10.7546/nntdm.2023.29.4.820-826
Full paper (PDF, 228 Kb)
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Authors and affiliations
Bakir Farhi 
Laboratoire de Mathématiques appliqueés,
Faculté des Sciences Exactes, Université de Bejaia
06000 Bejaia, Algeria
Abstract
In this note, based on a certain functional equation of the dilogarithm function, we establish nontrivial lower bounds for the p-adic valuation (where p is a given prime number) of some type of rational numbers involving harmonic numbers. Then we use our estimate to derive the integrality of some sequences of rational numbers, which cannot be seen directly from their definitions.
Keywords
- p-adic valuations
- Integer sequences
- Dilogarithm function
2020 Mathematics Subject Classification
- 11B83
- 11A41
- 33B30
References
- Dubickas, A. (in press). On a sequence of integers with unusual divisibility by a power of 2. Miskolc Mathematical Notes. Code: MMN-4276.
- Farhi, B. (2022). On a curious integer sequence. Preprint available at https://arxiv.org/abs/2204.10136.
- Farhi, B. (2022). The integrality of the Genocchi numbers obtained through a new identity and other results. Notes on Number Theory and Discrete Mathematics, 28(4), 749–757.
- Farhi, B. (2023). Nontrivial lower bounds for the p-adic valuation of certain rational numbers. Journal of Integer Sequences, 26, Article 23.6.1.
- Lewin, L. (1981). Polylogarithms and Associated Functions. Elsevier, New York.
- Moll, V. H. (2012). Numbers and Functions. Student Mathematical Library. American Mathematical Society, Providence.
- Zariski, O., & Samuel, P. (1960). Commutative Algebra. Vol II. Graduate Texts in Mathematics 29. Springer, New York.
Manuscript history
- Received: 12 April 2023
- Revised: 16 November 2023
- Accepted: 1 December 2023
- Online First: 23 December 2023
Copyright information
Ⓒ 2023 by the Author.
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
- Farhi, B. (2022). The integrality of the Genocchi numbers obtained through a new identity and other results. Notes on Number Theory and Discrete Mathematics, 28(4), 749–757.
Cite this paper
Farhi, B. (2023). Nontrivial lower bounds for the p-adic valuations of some type of rational numbers and an application for establishing the integrality of some rational sequences. Notes on Number Theory and Discrete Mathematics, 29(4), 820-826, DOI: 10.7546/nntdm.2023.29.4.820-826.
