Do Complex Primes Exist?

1. Mar 26, 2012

Dashin

Complex Numbers have always facinated me.

But... Do complex primes exist? If so, How?

2. Mar 26, 2012

Hurkyl

Staff Emeritus
There are no primes in the complex numbers. There are no primes in the real numbers either. When every non-zero number is invertible, there is no such thing as a prime!

In the integers, there are primes. There are primes in the gaussian integers as well. The gaussian integers are numbers of the form a + bi, where a and b are both integers.

In the gaussian integers, 2 is not prime; its prime factorization is (1-i)(1+i).

3. Mar 26, 2012

Dashin

Thanks!
So... What primes are there in the Gaussian Integers?

4. Mar 26, 2012

chiro

Not really but you could define an analog in terms of behavior. The thing you would need to think about are what the atoms are in the complex case and the constraints.

Did you have an idea of something in mind?

5. Mar 26, 2012

Dashin

6. Mar 26, 2012

Hurkyl

Staff Emeritus
If p is an integer prime, and you can write $a^2 + b^2 = |p|$, then both $a+bi$ and $a-bi$ are Gaussian integer primes. If you cannot, then $p$ is also a Gaussian integer prime.

All primes in the Gaussian integers are of this form.

7. Mar 26, 2012

Dashin

Does the order of a and b matter? Do you need to make the larger number be a, or doesn't it matter?

8. Mar 26, 2012

Hurkyl

Staff Emeritus
Doesn't matter; they just differ by a unit. Just like -5 and 5 are both integer primes -- and in a certain sense the "same" prime, (3+2i), i(3+2i), (-1)(3+2i), and (-i)(3+2i) are all the "same" gaussian integer prime.

The prime factorization of 2 I mentioned earlier: I could have (and probably should have) also written it as (-i) (1+i)^2, since 1+i and 1-i are the "same" prime.

(Just like prime factorizations in the integers can have a (-1) out front, factorizations in the guassian integers can have a (-1), (-i), or i out front)

If you don't like multiple primes being the "same", then I suppose you could insist on a being positive, and being larger in magnitude than b. (and have a special rule for deciding which of 1+i, 1-i, -1+i, and -1-i you like)

9. Mar 26, 2012

Dashin

Thank you very much.

10. Mar 26, 2012

Number Nine

There really aren't any good analogs in terms of behaviour. Every non-zero complex number is a unit and lacks a unique factorization, so even if your defined some sort of prime-analog, they would lack the importance of prime numbers in the integers (i.e. you can't really build anything with them).