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Re: [PrimeNumbers] Re: New Question

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  • Chris Caldwell
    ... The wonderful text: L. J. Mordell Diophantine Equations Academic Press, 1969, page 124-130 includes things like: Theorem 3: Let a = p or p^2 where p =
    Message 1 of 6 , Nov 1, 2003
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      At 01:12 PM 11/1/2003 +0000, you wrote:
      >I expect you're right, Chris (if you correct your typo and say X = -
      >r^n*Z ), and both the equation and your assertions have a familiar
      >ring to them, but I can neither prove that no other solutions exist
      >nor lay hands on a relevant reference.
      >Help us out, please!

      The wonderful text: L. J. Mordell "Diophantine Equations"
      Academic Press, 1969, page 124-130 includes things like:

      Theorem 3: Let a = p or p^2 where p = 2, 5 (mod 9) is a prime,
      and let e be a unit in Q(r) (r^2 + r + 1 = 0). Then the equation

      x^3 + y^3 + eaz^3 = 0

      has no solutions (x,y,z,e) in Q(r) except

      z = 0, x = -y, -ry, -r^2y

      unless a = 2, when the equation also has the solutions

      x^3 = y^3 = -e z^3, e = +/-1.

      Proof is about one page and is done by choosing x, y, z
      pairwise relatively prime and N(xyz) minimal. Then do
      descent...

      But we are wandering away from primes...

      Chris
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