Primes in the Gaussian integers
This picture by Oliver Knill
shows the prime numbers in the Gaussian integers. A Gaussian integer
is a complex number of the form a + bi, where a and b are integers and i is a square root of –1.
An important result about the ordinary integers is the Fundamental Theorem of Arithmetic
, which states that every integer greater than 1 can be expressed as a product of prime numbers in an essentially unique way. For example, the integer 21 can be written as 3 x 7, or 7 x 3, or (–3)x(–7), or (–7)x(–3). However, these factorizations are essentially the same, in the sense that they differ only in the order of the factors and by multiplication by +1 or –1. The numbers +1 and –1 are called units
, which means that they are the only integers z for which 1/z is also an integer.
An integer c can be defined to be prime
if it is not zero or a unit, but the only way to factorize c = ab is for either a or b to be a unit. If c is an integer bigger than 1, then this is equivalent to the familiar definition of prime, namely that the number has no factors other than 1 and itself. The negatives of prime numbers (such as –3 and –7, as above) also satisfy the definition of prime. However, the units +1 and –1 are not defined to be prime, because this would cause problems; for example, the Fundamental Theorem of Arithmetic would be false as stated.
It turns out, somewhat remarkably, that the analogue of the Fundamental Theorem of Arithmetic also holds for the Gaussian integers, which means that the Gaussian integers have a unique factorization property. It can be shown that the only units in the Gaussian integers are the numbers 1, i, –1 and –i; in other words, these are the only Gaussian integers z for which 1/z is also a Gaussian integer. Some primes in the ordinary sense, like 7 and 11, are also prime in the Gaussian integers. However, other primes, like 2 and 5, are not
prime when regarded as Gaussian integers: they can be factorized further as 2 = (1+i)(1–i) and 5 = (1+2i)(1–2i), and none of the factors is a unit. It is also possible to write 5 = (2–i)(2+i), but this is essentially the same as the previous factorization because the factors are unit multiples of the previous factors. (More precisely: we have 2–i = (–i)(1+2i) and 2+i = (i)(1–2i), where –i and i are both units.)
The picture plots the locations of the primes in the Gaussian integers. The prime numbers in the integers that remain prime in the Gaussian integers, such as 3, 7, 11 and 19, appear on the positive horizontal axis as the points (3,0), (7,0), (11,0) and (19,0). These are the prime numbers that congruent to 3 modulo 4; in other words, they leave a remainder of 3 when divided by 4. The entire picture is symmetric under rotation by a right angle, because a unit multiple of a prime is always a prime, and multiplying by the complex number i rotates the complex plane anticlockwise by a right angle.
In light brown, near the origin, we have the primes 1+i, 1–i, –1+i, and –1–i. These are the four primes that are factors of 2 = (1+i)(1–i). A knight's move from the origin, we have the eight primes (2+i), (1+2i), (–1+2i), (–2+i) and their negatives; these are the primes that are factors of 5, and we find the same behaviour for factors of a prime integer p whenever p is congruent to 1 modulo 4.
The Gaussian primes are closely related to the problem of writing a number as a sum of two squares. Pierre de Fermat
proved 1640 that if a prime is congruent to 1 mod 4, then it can always be written as a sum of two squares; for example, 13 = 9 + 4 = 3^2 + 2^2, and 29 = 25 + 4 = 5^2 + 2^2. (As usual, Fermat did not bother to provide a proof of this result; the first rigorous proof was provided by Euler in the 1740s.) The problem of factorizing ordinary primes in the Gaussian integers is essentially the same: 13 = (3+2i)(3–2i) and 29 = (5+2i)(5–2i).Relevant links
The picture appears in Knill's paper Goldbach for Gaussian, Hurwitz, Octavian and Eisenstein primes
), which, among other things, formulates a version of the Goldbach conjecture for Gaussian integers. Chapter 4 of the paper considers some more surprising questions, such as the frogger problem
, which asks what happens if we think of the picture as a scenario in the popular 1980s computer game Frogger
Another paper by Knill on a similar topic is the 71-page Some experiments in number theory
). Chapter 14 of that paper conjectures that if one runs Conway's Game of Life on the picture shown, then there is motion arbitrarily far from the origin. Knill has a website on these topics at http://math.harvard.edu/~knill/primes/
Wikipedia has more information about the Gaussian integers here: https://en.wikipedia.org/wiki/Gaussian_integer
Here's another post by me about the Gaussian integers in the work of my colleague Katherine Stange: https://plus.google.com/101584889282878921052/posts/eM3adto6nsj
I thank +Owen Maresh
for bringing some of these links to my attention in his post from June. I have been extremely busy for the last seven months, which is why I have barely posted, but this is no longer the case, so you can expect a decent level of output from me in the coming months.#mathematics #scienceeveryday