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Quotients of Hurwitz Primes

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.


Introduction
Quotient sets like {p/q : p, q are primes} have attracted the attention of mathematicians in the past three decades. It has been proved (or observed) many times that the set {p/q : p, q are primes} is dense in the positive real numbers (e.g., [2,Exercise 218], [5,Corollary 5], [6,Theorem 4] ). In 2013, Garcia [3] considered the set of all quotients of Gaussian primes and proved that it is dense in the complex plane. Later Garcia and Luca [4] proved that the set of quotients of nonzero Fibonacci numbers is dense in the p-adic numbers for every prime p. Sanna [8] generalized Garcia and Luca's result and proved that for any integer k ≥ 2 and any prime number p, the set of quotients of nonzero k-generalized Fibonacci numbers is dense in the p-adic numbers.
Recently, Sittinger [9] 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. Sittinger also asked an interesting open question in his paper: Is the set of quotients of Hurwitz primes dense in the quaternions (see Section 2 for the definitions)? In this paper, we answer Sittinger's question and prove the following theorem. Remark 1.1. As the multiplication of quaternions is not commutative, for any two non-zero quaternions a, b, their quotient could be defined by b a = ba −1 or b a = a −1 b and Theorem 1.1 holds for both cases. In this paper, we only prove the former case and the proof of the latter case is very similar with obvious modifications.
Remark 1.2. In fact we prove a slightly stronger result than Theorem 1.1. Hurwitz quaternions could be divided into two disjoint subsets (see Section 2 for the notation and definitions) Our proof indicates that the set p q : p and q are Hurwitz primes in H 1 is dense in the quaternions. Moreover, for any two Hurwitz primes p, q ∈ H 1 with odd norms, it is easy to see that pu, qu are Hurwitz primes belonging to H 2 and pu qu = (pu)(qu) −1 = p q , where u = 1+i+j+k 2 is a unit in Hurwitz quaternions. Therefore, the set p q : p and q are Hurwitz primes in H 2 is also dense in the quaternions.

Hurwitz quaternions
In this section, we introduce some properties of quaternions and most of the materials can be found in [1].
The quaternions were discovered by Irish mathematician Hamilton in 1843. They have been widely used in the electrodynamics, general relativity, navigation, satellite attitude control and other fields.
where i, j, k commute with every real number and satisfy Let a = a 1 + a 2 i + a 3 j + a 4 k and b = b 1 + b 2 i + b 3 j + b 4 k be any two quaternions. The addition of quaternions is defined by For any real number λ, the scalar multiplication is defined by Then the quaternions form a vector space with these two operations. Moreover, we can define the multiplication of quaternions by Clearly, we have ij = k and ji = −k so the multiplication of quaternions is not commutative. For It is easy to see that Definition 2.4. For any a = a 1 + a 2 i + a 3 j + a 4 k ∈ Q and a = 0, its inverse is defined by In this paper, the quotient of two quaternions is defined by b a = ba −1 .
One interesting subset of quaternions is the set of Hurwitz quaternions which was introduced by Hurwitz in 1919. 3 Definition 2.5. The set of Hurwitz quaternions H is a subset of quaternions, defined as We say that a is a unit in H if a = 1.
It is easy to see that for any a ∈ H, a ∈ Z. Moreover, we have the following result. Definition 2.7. We say that p ∈ H is a Hurwitz prime if p is not zero or a unit and is not a product of non-units in H.
We have the following result to determine whether a Hurwitz quaternion is a Hurwitz prime.

Preliminaries
In this section, we introduce some tools which will be used later. We begin with some well-known properties of R 4 . For any two vectors − → x = (x 1 , x 2 , x 3 , x 4 ), − → y = (y 1 , y 2 , y 3 , y 4 ) ∈ R 4 , the metric is defined by Define a map σ from Q to R 4 by σ : Then it is easy to see that σ is an isomorphism. Moreover, σ is also an isometry and for any a, b ∈ Q. Next, we introduce our main tool. Denote by S the four dimensional hypersphere x 2 1 + x 2 2 + x 2 3 + x 2 4 = 1. For any 0 < θ < π and − → x ∈ R 4 , define is a hyperspherical cap in S and denote by A(Ω( − → x , θ)) its surface area. Clearly A(Ω( − → x , θ)) is a positive real number and only depends on θ and − → x . Define The following theorem is a special case of [7, Theorem 1] with Q(X) = x 2 1 + x 2 2 + x 2 3 + x 2 4 and Ω = Ω( − → x , θ). Theorem 3.1. Let notation be as above. For any positive integer n with (n, 2) = 1 and ε > 0, we have where r(n) is the number of integral solutions of x 2 1 + x 2 2 + x 2 3 + x 2 4 = n and A(S) is the surface area of S. It is sufficient to prove that for any quaternion h and any ε > 0, there exist two Hurwitz primes p, q such that We first consider the case h = 0. Since the set of all quotients of prime numbers is dense in positive real numbers, there exist two prime numbers p, q such that p/q < ε 2 . By Lemma 2.6 and Lemma 2.8, there exist two Hurwitz primes p, q such that p = p, q = q and In what follows, we assume h = 0 and without loss of generality, we assume By Theorem 3.1 and (3.4), for any positive odd integer n, we have is positive and only depends on ε and h, there exists if n > N 1 . By similar arguments, there exists N 2 = N 2 (ε 1 , − → e 1 ) such that r(n, Ω( − → e 1 , ε 1 )) > 1 if n > N 2 , where − → e 1 = (1, 0, 0, 0). Moreover, by the Prime Number Theorem, there exists N 3 = N 3 (ε, h) such that the interval (n( h − ε 2 /10), n( h + ε 2 /10)) contains at least one prime number if n > N 3 . Let q be a prime number satisfying Then (4.3) r(q, Ω( − → e 1 , ε 1 )) > 1 and there exists a prime (4.4) p ∈ (q( h − ε 2 /10), q( h + ε 2 /10)).

Acknowledgement
It is our pleasure to thank professor Yingnan Wang who is from Shenzhen Univeristy for his help and helpful advice throughout this project.