## Re: [PrimeNumbers] Re: Complex primes?

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• Isn t i+1 = i*(1-i) and so not prime? rob ... From: Mike Oakes To: primenumbers@yahoogroups.com Sent: Tuesday, March 20, 2007 12:49 AM Subject: [PrimeNumbers]
Message 1 of 5 , Mar 20, 2007
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Isn't i+1 = i*(1-i) and so not prime?

rob

----- Original Message -----
From: Mike Oakes
Sent: Tuesday, March 20, 2007 12:49 AM

--- 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]
• ... 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
Message 2 of 5 , Mar 20, 2007
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On 3/20/07, Rob <robdine@...> wrote:
> Isn't i+1 = i*(1-i) and so not prime?
>
> rob

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.

--Joshua Zucker
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