### Gaussian Primes

We are all familiar with splitting natural numbers into prime components. This decomposition is unique, except for the order of the factors. We can apply the idea of prime components to many more general sets of numbers.

The Gaussian integers are all the complex numbers with integer real and imaginary parts, that is, all numbers in the set

$\displaystyle \mathbb{Z}[i] \equiv \{ m + i n : m, n \in \mathbb{Z} \} \,.$

The set ${\mathbb{Z}[i]}$ forms a two-dimensional lattice in the complex plane. For any element ${g \in \mathbb{Z}[i]}$ we consider the four numbers ${\{g, -g, ig, -ig \}}$ as associates. The associates of ${1}$ are known as units: ${\{1, -1, i, -i \}}$.

Unique Factorization Theorem

A Gaussian prime is an element of ${\mathbb{Z}[i]}$ that cannot be expressed as a product of non-unit Gaussian integers. There is a unique factorization theorem for ${\mathbb{Z}[i]}$: every Gaussian integer can be factored uniquely as a product of a unit and of Gaussian primes, unique up to replacement of any Gaussian prime by any of its associates and change of the unit.

Why the long-winded language here? We see that the number ${2}$ can be expressed as

$\displaystyle 2 = (1 + i)(1 - i) = i(-i + 1)(1 - i) = i(1 - i)^2 \,.$

Note that ${1+i}$ and ${1-i}$ are associates, which are considered as equivalent factors.

Obviously, no number composite in ${\mathbb{N}}$ is a Gaussian prime. But what is less obvious is that not every prime in ${\mathbb{N}}$ is a Gaussian prime. Let us look at how some small prime numbers may be split into factors in ${\mathbb{Z}[i]}$:

$\displaystyle 5 = (2 + i)(2 - i) \qquad 13 = (3 + 2i)(3 - 2i) \qquad 17 = (4 + i)(4 - i) \,.$

Note that all these are primes of the form ${4k+1}$. This is no accident: Fermat’s Christmas Theorem ensures us that a prime may be expressed as the sum of two squares if and only if it is congruent to 1 modulo 4. But the sum of two squares can be expressed as the product of two complex conjugate Gaussian integers, ${a+ib}$ and ${a-ib}$:

$\displaystyle (a+ib)(a-ib) = a^2 + b^2 \,.$

Thus, a prime of the form ${4k+1}$ cannot be a Gaussian prime. On the other hand, every prime of the form ${4k+3}$ is also a Gaussian prime.

Gaussian Primes in the Complex Plane

In the figure below we plot the Gaussian primes ${a+ib}$ for ${|a|\le 25}$ and ${|b|\le 25}$ (left panel) and for ${|a|\le 50}$ and ${|b|\le 50}$ (right panel).

Gaussian primes ${a+ib}$ for ${|a|}$ and ${|b|}$ less than 25 (left) and less than 50 (right).

Two Unsolved Problems

I. Gauss Circle Problem

How many Gaussian integers, or unit lattice points, are contained within (or on) a circle of radius ${r}$ centered at the origin? This is equivalent to asking how many numbers ${m +in}$ with ${m,n\in\mathbb{N}}$ are contained in the closed disk ${m^2+n^2 \le r^2}$?

Obviously, each generic unit square aligned with the lattice contains a single lattice point, so the number of points in the circle should be approximately given by the area ${\pi r^2}$. The challenge is to bound the error term. Gauss made some strides in solving the problem, but a general solution is lacking.

II. Gaussian Moat Problem

Another number theory problem: is it possible to find an infinite sequence of Gaussian primes such that the distance between each consecutive pair is bounded above by some fixed number. The problem is often expressed in terms of finding a route in the complex plane from the origin to infinity, using the primes in the figure above as stepping-stones, with the length of each step never exceeding some uniform bound. This problem remains unsolved.

Sources

${\bullet}$ Fraleigh, John B., 1970: A First Course in Abstract Algebra. Addison-Wesley Pub. Co., 447 pp. [See Chapter 34.].

${\bullet}$ Knill, Oliver (Harvard): Some beautiful images of Gaussian primes can be found on this website.