int. electron. j. geom. International Electronic Journal of Geometry 1307-5624 Kazım İlarslan 10.36890/iejg.1759957 Algebraic and Differential Geometry Pure Mathematics (Other) Cebirsel ve Diferansiyel Geometri Temel Matematik (Diğer) The Number of $k$-potent Elements in the Quaternion Algebra $\mathbb{H}_{\mathbb{Z}_{p}}$ https://orcid.org/0000-0003-2714-0583 Flaut Cristina Ovidius University https://orcid.org/0009-0004-8162-6212 Baias Andreea Ovidius University of Constanta, Romania 04 22 2026 19 1 216 228 08 07 2025 10 28 2025 Copyright © 2008, International Electronic Journal of Geometry 2008 International Electronic Journal of Geometry

In this paper we count the number of $k$ -potent elements over $\mathbb{H}_{\mathbb{Z}_{p}}$ ,where $\mathbb{H}_{\mathbb{Z}_{p}}$ is the quaternion algebra over $\mathbb{Z}_{p}$ , and we present a descriptive formula for thegeneral case. For $k\in \{3,4,5\}$ , we give an explicit formula forthese values. Moreover, as an application of these results, we count thenumber of solutions of the equation $x^{k}=1$ over $\mathbb{H}_{\mathbb{Z}_{p}}$. For this purpose, we will use computer as a toolto check and understand the behavior of these elements in all cases that will be studied.

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