- Isn't i+1 = i*(1-i) and so not prime?

rob

----- Original Message -----

From: Mike Oakes

To: primenumbers@yahoogroups.com

Sent: Tuesday, March 20, 2007 12:49 AM

Subject: [PrimeNumbers] Re: Complex primes?

--- In primenumbers@yahoogroups.com, peter piper <terranorca@...> wrote:

>

> I apologize for the naivete of my question, but I am

> not a mathematician.

>

> Having read a few books on Riemann and prime numbers,

> I have this question:

>

> Does Riemann's extension of the zeta function to the

> complex plane imply that there are complex prime

> numbers?

>

> I have seen lists of prime numbers and lists of zeta

> zeros, but not of complex primes. Indeed, I don't even

> know if the idea of complex prime makes any sense.

>

It makes perfect sense, and they are often called "Gaussian primes".

If you enter that search term into google, you will find lots of useful

introductory articles.

The simplest example of a Gaussian prime is 1+i; this is prime because,

as you can readily verify, there are no other complex integers whose

product (a1 + i*b1)*(a2 + i*b2) = (1+i).

This is a fascinating subject - but it has nothing at all to do with

Riemann's zeta function.

Hope this helps.

-Mike Oakes

[Non-text portions of this message have been removed] - On 3/20/07, Rob <robdine@...> wrote:
> Isn't i+1 = i*(1-i) and so not prime?

Hi Rob,

>

> rob

that's analogous to saying "Isn't 7 = -1 * -7 and so not prime?"

The rules for primes are no divisors except for UNITS and themselves,

where units are things that have reciprocals. Since 1/i = -i is also

a Gaussian integer, it's a unit and so we don't count it.

--Joshua Zucker