Let’s go back a bit!

- Natural Numbers N --

First we have the natural numbers N: 1, 2, 3, 4, etc. This system is closed under addition and multiplication, but not subtraction.

- Integers Z --

To make the system Z that is also closed under subtraction we add to N the integers 0, −1, −2, −3, −4, etc.

- Imaginary integers --

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 0*i*, +/−1*i*,
+/−2*i*, +/−3*i*, +/−4*i*, etc.

- Gaussian Integers Z[
*i*] --

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*

- Gaussian Primes --

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 +
2*i*) * (1 − 2*i*) = (2 + *i*) * (2 −*i*). The
number 5 has 8 Gaussian prime integer divisors 1 + 2*i*, 1 −
2*i*, −1 + 2*i*, −1 − 2*i*, 2 + *i*, 2
− *i*, −2 + *i*, and −2 − *i*.

In general, prime integer of the form |*p*| = 4*k* + 1 (5, 13, 17, 29,
etc.) are not Gaussian primes and prime integer of the form |*p*| =
4*k* + 3 (3, 7, 11, 19, 23, etc.) are Gaussian primes.

Prime integers of the form |*p*| = 4*k* + 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* = 4*k* + 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* = 4*k* + 1 = 5, 13, 17,
29, etc.

Return to Gaussian Primes

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This page accessed times since October 20, 2004.

Page created by: hjsmithh@sbcglobal.net

Changes last made on Monday, 18-Jul-05 12:18:00 PDT