Title: Equivalent of prime numbers in the complex plane?
Post by: cbuchner1 on December 25, 2009, 02:06:04 PM
Hi everybody.
Is there an equivalent to prime numbers in the complex plane? (and i do not mean the primes along the real axis, hehe)
Is there an equivalent to prime numbers in the complex plane? (and i do not mean the primes along the real axis, hehe)
Title: Re: Equivalent of prime numbers in the complex plane?
Post by: kram1032 on December 25, 2009, 02:48:51 PM
Well just thinking but:
Primes are defined to be numbers which can't be split by any positive Integer to form an other one, except for 1 and the number itself.
So, basically, primes "don't care" about sign.
Which would be my guess that the closest equivalent would be kind of a signed prime.
)
with p being a prime and phi being any angle...
Though that most likely is not what you search for, right?
An other variation would be to extend it to Gaussean or Eisenstein Integers but that most likely would lead to too many values. Dunno...
It would be that you can't divide any complex prime p with signed Integers in the arguments without loosing the integers if you don't use 1, -1, sgn(p) or p itself....
But I'm not sure if there are other conditions which you'd need for this...
Or if it actually would make sense to define something like this.
\over a^2+b^2)
So, "simply" test wether
returns an integer or not and wether 
does.
if one of them is true, you don't have a prime.
Most likely, there is a better way to test it. Didn't really look into searching for primes
Primes are defined to be numbers which can't be split by any positive Integer to form an other one, except for 1 and the number itself.
So, basically, primes "don't care" about sign.
Which would be my guess that the closest equivalent would be kind of a signed prime.
Though that most likely is not what you search for, right?
An other variation would be to extend it to Gaussean or Eisenstein Integers but that most likely would lead to too many values. Dunno...
It would be that you can't divide any complex prime p with signed Integers in the arguments without loosing the integers if you don't use 1, -1, sgn(p) or p itself....
But I'm not sure if there are other conditions which you'd need for this...
Or if it actually would make sense to define something like this.
So, "simply" test wether
if one of them is true, you don't have a prime.
Most likely, there is a better way to test it. Didn't really look into searching for primes
Title: Re: Equivalent of prime numbers in the complex plane?
Post by: BradC on December 25, 2009, 06:24:34 PM
There are these: http://mathworld.wolfram.com/GaussianPrime.html (http://mathworld.wolfram.com/GaussianPrime.html)
Title: Re: Equivalent of prime numbers in the complex plane?
Post by: cbuchner1 on December 25, 2009, 07:22:43 PM
There are these: http://mathworld.wolfram.com/GaussianPrime.html (http://mathworld.wolfram.com/GaussianPrime.html)
Sure looks like these numbers form patterns in the complex plane. Maybe not enough to be called fractal, but still...
Wikipedia has a pretty large plot: http://en.wikipedia.org/wiki/File:Gauss-primes-768x768.png
And here is a JAVA applet plotting them interactively: http://www.alpertron.com.ar/GAUSSPR.HTM