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Re: [PrimeNumbers] Re: Complex primes?

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  • Rob
    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
      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]
    • Joshua Zucker
      ... 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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