The number of k-potent elements in the quaternion algebra HZp

TL;DR

This paper counts the number of k-potent elements in the quaternion algebra over Zp, providing explicit formulas for k in {3, 4, 5}, and applies these results to solve related equations.

Contribution

It introduces explicit formulas for counting k-potent elements in quaternion algebras over Zp, extending previous understanding in algebraic structures.

Findings

Explicit formulas for k in {3, 4, 5}

Counts solutions to x^k = 1 in HZp

Descriptive formula for general k

Abstract

In this paper we count the number of k-potent elements over HZp , the quaternion algebra over Zp, and we give a descriptive formula for the general case. For k in {3, 4, 5}, we give an explicit formula for these values. Moreover, as an application, we count the number of solutions of the equation xk = 1 over HZp .