In this paper we count the number of k -potent elements over Hₙ_, where Hₙ_ is the quaternion algebra over Z, and we present a descriptive formula for thegeneral case. For k \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 Hₙ_. 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.
Flaut et al. (Wed,) studied this question.