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Re: Complex a*x^n+b*y^n puzzle

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  • mikeoakes2
    ... Nice suggestion, David. Must get coding... (Of course the train of thought then goes:- or Eisenstein integers, or algebraic integers of the ring
    Message 1 of 37 , Nov 28, 2009
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      --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
      >
      > --- In primenumbers@yahoogroups.com,
      > "mikeoakes2" <mikeoakes2@> wrote:
      >
      > > I do think x or y should be allowed to be negative - why not?
      >
      > I think they should be allowed to be complex - why not?
      >
      > Complex a*x^n+b*y^n puzzle: Find 4 Gaussian integers a,b,x,y
      > such that x and y are not units and a*x^n+b*y^n is a Gaussian
      > prime, for 1 <= n <= n_max, with n_max as large as possible.

      Nice suggestion, David.
      Must get coding...

      (Of course the train of thought then goes:-
      or Eisenstein integers,
      or algebraic integers of the ring k(sqrt(m)),
      or... :-)

      Mike
    • djbroadhurst
      ... There is only rational prime of the form a^2 + b^2 that yields precisely 4 distinct Gaussian primes, namely 2 = 1^2 + 1^2. If a^2 + b^2 is an odd rational
      Message 37 of 37 , Dec 5, 2009
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        --- In primenumbers@yahoogroups.com,
        "Robdine" <robdine@...> wrote:

        > any rational prime that can be represented by the sum
        > of 2 squares (a^2+b^2) will define 4 gaussian primes

        There is only rational prime of the form a^2 + b^2 that yields
        precisely 4 distinct Gaussian primes, namely 2 = 1^2 + 1^2.

        If a^2 + b^2 is an odd rational prime, we have
        8 [sic] asociates of the Gaussian prime z = a + I*b,
        since we may mulitply it and its conjugate z = a - I*b
        by the 4 units I, -I, -1, 1, obtaining 8 distinct
        Gaussian integers that are prime.

        David
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