Let’s go back a bit!
First we have the natural numbers N: 1, 2, 3, 4, etc. This system is closed under addition and multiplication, but not subtraction.
To make the system Z that is also closed under subtraction we add to N the integers 0, −1, −2, −3, −4, etc.
We can see that some positive integers are equal to the square of some other integers. For example, 4 = 2 * 2 and 4 = (−2) * (−2), but no negative integer is the square of an integer. So we define a new number i such that i^2 = −1. The imaginary integers are 0i, +/−1i, +/−2i, +/−3i, +/−4i, etc.
If you start doing arithmetic with imaginary integers you quickly realize that you need a system that has more than integers and imaginary integers. For example, if we try to evaluate i * (−i) + i, which contains nothing but imaginary integers, we get 1 + i that is neither an integer nor an imaginary integer. We call it a Gaussian integer. The Gaussian integers Z[i] are the numbers a + bi where a and b are integers. We call a the real part and b the imaginary part.
Gaussian integers are closed under addition:
(a + bi) + (c + di) = (a + c) + (b + d)i
Gaussian integers are closed under multiplication:
(a + bi) * (c + di) = (a * c − b * d) + (a * d + b * c)i
A Gaussian prime is a Gaussian integer p with exactly 8 divisors: p, −p, pi, −pi, 1, −1, i, and −i. The numbers 1, −1, i, and −i are the Gaussian units; they divide every number in Z[i].
For example, the prime integer 2 is not a Gaussian prime because 2 can be written as (1 + i) * (1 − i). The number 2 has 4 Gaussian prime divisors 1 + i, 1 − i, −1 + i, and −1 − i.
The prime integer 5 is not a Gaussian prime because 5 can be written as (1 + 2i) * (1 − 2i) = (2 + i) * (2 −i). The number 5 has 8 Gaussian prime integer divisors 1 + 2i, 1 − 2i, −1 + 2i, −1 − 2i, 2 + i, 2 − i, −2 + i, and −2 − i.
In general, prime integer of the form |p| = 4k + 1 (5, 13, 17, 29, etc.) are not Gaussian primes and prime integer of the form |p| = 4k + 3 (3, 7, 11, 19, 23, etc.) are Gaussian primes.
Prime integers of the form |p| = 4k + 1 have a very special property: They can be decomposed into the sum of two squares p = m^2 + n^2 (m < n) in one and only one way. For example, 5 = 1^2 + 2^2, 13 = 2^2 + 3^2, 17 = 1^2 + 4^2, 29 = 2^2 + 5^2.
Gaussian primes are the Gaussian integers that are in the following 3 classes:
(a) The 4 non-trivial divisors of 2: 1 + i, 1 − i, −1 + i, and −1 − i.
(b) The 4 trivial divisors of each of the prime integers > 2 that cannot be factored even in Z[i]: p, −p, pi, and −pi, where p = 4k + 3 = 3, 7, 11, 19, 23, etc.
(c) The 8 non-trivial divisors of each of the prime integers > 2 that can be factored in Z[i]: m + ni, m − ni, −m + ni, −m – ni, n + mi, n − mi, −n + mi, and −n − mi, where m^2 + n^2 = p = 4k + 1 = 5, 13, 17, 29, etc.
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October 20, 2004.