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Re: Complex primes?

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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 1 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 2 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 3 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 4 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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