Quotients of Gaussian Primes

It has been observed many times, both in the Monthly and elsewhere, that the set of all quotients of prime numbers is dense in the positive real numbers. In this short note we answer the related question: “Is the set of all quotients of Gaussian primes dense in the complex plane?” Quotient sets {s/t : s, t ∈ S} corresponding to subsets S of the natural numbers have been intensely studied in the Monthly over the years [1,4,7,8,10,13]. Moreover, it has been observed many times in the Monthly and elsewhere that the set of all quotients of prime numbers is dense in the positive reals (e.g., [2, Ex. 218], [3, Ex. 4.19], [8, Thm. 4], [4, Cor. 5], [11, Ex. 7, p. 107], [12, Thm. 4], [13, Cor. 2]). In this short note we answer the related question: “Is the set of all quotients of Gaussian primes dense in the complex plane?” The author became convinced of the nontriviality of this problem after consulting several respected number theorists who each admitted not seeing a simple solution. In the following, we refer to the traditional primes 2, 3, 5, 7, . . . as rational primes, remarking that a rational prime p is a Gaussian prime (i.e., a prime in the ring Z[i] := {a + bi : a, b ∈ Z} of Gaussian integers) if and only if p ≡ 3 (mod4). In general, a nonzero Gaussian integer is prime if and only if it is of the form ±p or ±pi where p is a rational prime congruent to 3 (mod4) or if it is of the form a+ bi where a + b is a rational prime (see Figure 1). We refer the reader to [5] for complete details. Theorem. The set of quotients of Gaussian primes is dense in the complex plane. Figure 1. Gaussian primes a + bi satisfying |a|, |b| ≤ 50 and |a|, |b| ≤ 100, respectively.