The Number of $k$-potent Elements in the Quaternion Algebra $\mathbb{H}_{\mathbb{Z}_{p}}$

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.