Re: [PrimeNumbers] Re: Complex primes?
- Isn't i+1 = i*(1-i) and so not prime?
----- Original Message -----
From: Mike Oakes
Sent: Tuesday, March 20, 2007 12:49 AM
Subject: [PrimeNumbers] Re: Complex primes?
--- In firstname.lastname@example.org, 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
> 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
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.
[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,
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.