About 10 years ago, I latched onto the concept of Gaussian Primes.
A Complex (or Gaussian) Prime works much the same way that a real-valued prime does. It has no divisors other itself and the unit element.
In the real numbers, the units are 1 and -1. In the Complex number system, the units are 1, -1, i, -i.
In order to tell if a complex number (a+bi) is prime, simply compute a²+b². If the result is a real-valued prime, then the original complex number (a+bi) is prime. The proof of this is actually pretty simple and interesting.
Today, in Intermediate Algebra, we were caculating some multiplication problems involving complex numbers.
One of these was (5-3i)(1+i)=8+2i
I noticed that the answer was factorable as 2(4+i) and then realized that these were two different factor pairs for the complex number 8+2i. That meant that I should be able to turn one factor pair into the other by shifting around the prime factors of the original number.
An example of this is that 24=6*4 and 24=3*8, so I can create one factor pair from the other by shifting a factor of 2.
In considering 6*4, if we look at the 6 as 3*2, then 6*4=(3*2)*4=3*(2*4)=3*8.
Pretty simple, in the real number system.
In the example we looked at in class today we said that
Now, I knew that 1+i is prime (because 1²+1²=2 which is prime) and I suspected that the 2 would factor into (1+i)(1-i), which would mean that 5+3i=(4+i)(1-i), which it does.
So the prime factorization of 8+2i is (1+i)(1-i)(4+i).
Each of these factors is prime and can be checked using the little trick of a²+b²=P.
Here’s a second problem, I’ll include the factorization below…
Find the prime factorization of 4+32i given that:
I mentioned earlier that the proof for this is simple and interesting. I’ll include it below as well…
PROOF for primality test:
So, we say that if a²+b²=P, then a+bi is prime and we’ll prove this by contradiction.
Assume that a²+b²=P, but that a+bi is NOT prime, or that
This means that
Which contradicts our assumption that a²+b²=P.
So if a²+b²=P, then a+bi is a complex prime.