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Complex primes?

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  • peter piper
    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
    Message 1 of 5 , Mar 19, 2007
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      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.

      Thank you for indulging my curiosity.



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    • Mike Oakes
      ... 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
      Message 2 of 5 , Mar 19, 2007
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        --- 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
      • Phil Carmody
        ... Not really, or not directly. That doesn t mean there s no such thing as complex primes - to the contrary, they re a fairly well studied field, it s just
        Message 3 of 5 , Mar 19, 2007
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          --- 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?

          Not really, or not directly. That doesn't mean there's no such
          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
          > zeros, but not of complex primes. Indeed, I don't even
          > know if the idea of complex prime makes any sense.

          If you've got a ring, then you've got the concept of both primes and
          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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        • 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 4 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 5 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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