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© Copyright 1999, Jim Loy
A prime number (such as 2 or 3 or 5) is a natural number (positive whole number) greater than 1, which is only divisible by itself and 1 (and no other natural number). The other natural numbers (greater than one), are called composite numbers, and are the products of prime numbers. One is neither prime nor composite. See my articles, The Infinitude Of Primes and Sieve Of Eratosthenes.
In the study of complex numbers, there are similar prime numbers, called Gaussian primes. The following paragraph is from my article Imaginary Numbers. It will introduce us to complex numbers.
An imaginary number is a real number times the positive square root of -1. The positive square root of -1 is usually called i. It is not an integer, it is not real. It is the answer to the question, "What number squared is -1?" Well, no normal number can ever be the answer to that question. But, there are good reasons for defining a new kind of number, imaginary numbers, just to answer that question. 3i or -7i are imaginary numbers. Actually, if we define imaginary numbers, we also have to define complex numbers. A complex number is a real number plus an imaginary number. An imaginary number is really a complex number, like 0+3i.
And a regular integer, like 5, is also a complex number: 5+0i. A Gaussian integer is a complex number, with integers for both its real part and its imaginary part. A Gaussian prime is a Gaussian integer which is only divisible by itself and 1 (and no other Gaussian integer). We will only examine complex numbers in which the real part is positive and is greater than or equal to the absolute value of the imaginary part. When we begin to experiment with such primes, we quickly learn that these Gaussian primes come in pairs, a+bi and a-bi. These both fit the above definition as a Gaussian integer. And, when we multiply them together, we get an integer: (a+bi)(a-bi)=a2+b2. So, some normal primes (the integer type) may be the product of Gaussian primes. In other words, these integer primes may not be Gaussian primes. Others may be. Let's examine that situation.
Taking a clue from the Sieve Of Eratosthenes, we will start with the smallest integer complex number 1+i and assume that this is a Gaussian prime, and see what integer we get:
(1+i)(1-i)=2
So 2 is not a Gaussian prime. The next one is 2+i (We are going in order of the size of the product, here):
(2+i)(2-i)=5
1+2i would give us the same answer. That is why I stipulated that we would only examine complex numbers in which the real part is >=the imaginary part. So our primes come in sets of four. Here they are 2+i, 2-i, 1+2i, and 1-2i. We have skipped over 3, which would seem to be a Gaussian prime. We can also skip 2+2i, which is obviously composite. Next we try 3+i:
(3+i)(3-i)=10
10 is composite. So we can deduce that 3+i is also composite (in the Gaussian prime sense), apparently being divisible by 1+i. Dividing it out, we find that (3+i)/(1+i)=2-i. We can deduce that a+bi is not prime if both a and b are odd (except for 1+i), because then a2+b2 will be even. Now we try 3+2i:
(3+2i)(3-2i)=13
3+2i would seem to be a Gaussian prime, and 13 is not. Primes we have skipped over are 7 and 11, which would seem to be Gaussian primes. 4+i is next:
(4+i)(4-i)=17
So 4+i is a prime, and 17 is not. Next, 3+3i and 4+2i are obviously composite. 4+3i is next:
(4+3i)(4-3i)=25
25 was already composite, so 4+3i is divisible by 2+i. 5+i is composite, as both 5 and 1 are odd. 5+2i is next:
(5+2i)(5-2i)=29
5+3i is composite. 6+i is next:
(6+i)(6-i)=37
(5+4i)(5-4i)=41
(7+2i)(7-2i)=53
(6+5i)(6-5i)=61
(7+4i)(7-4i)=65 (composite)
(8+i)(8-i)=65 (composite)
(8+3i)(8-3i)=73
(7+6i)(7-6i)=85 (composite)
(9+2i)(9-2i)=85 (composite)
(8+5i)(8-5i)=89
(9+4i)(9-4i)=97
Let's list the Gaussian primes that we have found, including the integer primes that we skipped over:
Here are the integers < 100 which are Gaussian primes:
See Mathematica in Education and Research (a program to plot Gaussian Primes on the complex plane). Also Ask Dr. Math (13 is not a Gaussian Prime, but 3 is)