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

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

> 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.

>

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 - --- peter piper <terranorca@...> wrote:
> I apologize for the naivete of my question, but I am

Not really, or not directly. That doesn't mean there's no such

> 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?

thing as complex primes - to the contrary, they're a fairly well

studied field, it's just that it's not the extension of the Riemann

zeta function to the whole complex plane that brings them into

existance. Their study predates Riemann by quite a way.

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

If you've got a ring, then you've got the concept of both primes and

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

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

irreducible elements (which are what the layman calls primes, as in most

situations, the irreducibles are precisely the primes). That's

_any_ ring.

The most common 'complex' primes are the primes in the Gaussian Integers,

i.e. x+iy, where i=sqrt(-1), x, y integers. There's a plot of them on

mathworld - quite pretty in their symmetry.

My personal faourites are the primes in the Eisenstein Integers, again,

there's a plot on mathworld.

For these two examples, there's a direct correspondence between the usual

('rational') primes and complex primes. (Either a rational prime remains prime

or it splits into precisely two new complex conjugate primes.)

There are an infinitude of such rings; many share similar properties but many

variations are possible.

Phil

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http://autos.yahoo.com/green_center/ - 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