Counts $k$-potent elements in quaternion algebra, revealing insights into their properties and solutions to equations.
In this paper we count the number of k -potent elements over H_Zₚ ,where H_Zₚ is the quaternion algebra over Zₚ , 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 xᵏ=1 over H_Zₚ. 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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Flaut et al. (2026) studied this question.
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