# Quotients of Hurwitz Primes

## TL;DR

This paper proves that the set of all quotients of Hurwitz primes is dense in the quaternions, extending the understanding of prime quotient density to quaternionic numbers.

## Contribution

It establishes that quotients of Hurwitz primes are dense in the quaternions, answering a previously open question in number theory.

## Key findings

- Quotients of Hurwitz primes are dense in the quaternions.
- Extends density results from real and complex primes to quaternionic primes.

## Abstract

Quotient sets have attracted the attention of mathematicians in the past three decades. The set of quotients of primes is dense in the positive real numbers and the set of all quotients of Gaussian primes is also dense in the complex plane. Sittinger has proved that the set of quotients of primes in an imaginary quadratic ring is dense in the complex plane and the set of quotients of primes in a real quadratic number ring is dense in R. An interesting open question is introduced by Sittinger: Is the set of quotients of Hurwitz primes dense in the quaternions? In this paper, we answer the question and prove that the set of all quotients of Hurwitz primes is dense in the quaternions.