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24381Re: Impossible to Prove ??

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  • djbroadhurst
    Aug 11, 2012
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      "Aldrich" <aldrich617@...> wrote:

      > I don't see how
      > it immediately follows from Hardy/Wright Theorem 257 that
      > A = 5x^2 + 5xy + y^2 has as values ALL of the possible
      > values of primes that are +/- 1 mod 10.

      Let w = (1+sqrt(5))/2 be the fundamental unit.
      HW prove that every prime p = +/-1 mod 10 is of the form
      norm(a+b*w) = a^2 + a*b - b^2
      for some integer pair [a,b].

      Lemma: Every prime p = +/-1 mod 10 is of the form
      norm(2*x+y+x*w) = 5*x^2 + 5*x*y + y^2
      for some integer pair [x,y].

      Proof: Set x = b, y = a-2*b.

      David
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