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Re: primes and squares

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  • Mark Underwood
    Thank you Decio and Dario Decio I looked up Fermat s 4n+1 Theorem and it is as you say, and proved by Euler. (That is, every prime of the form 4n+1 can be
    Message 1 of 6 , Apr 4, 2005
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      Thank you Decio and Dario

      Decio I looked up Fermat's "4n+1" Theorem and it is as you say, and
      proved by Euler. (That is, every prime of the form 4n+1 can be
      uniquely expressed as x^2 + y^2.) Thanks for the input on the other
      forms. I have a lot to learn.

      Dario, your data was great, thanks. Because of it I am going to
      investigate something further. I had only done solutions counts, not
      investigated the properties of the actual solutions before.

      Your data showed many solutions, and I said their was only one
      solution for p = x^2 -2*y^2 , p = 8m+7. What gives?

      Brings out another one of my mistakes! I had said that y<p/2, but
      what I meant to say was that y < p^2 /2. In such a case there is only
      one solution.

      I've book marked your very cool site BTW!

      Mark








      --- In primenumbers@yahoogroups.com, "Dario Alpern" <alpertron@h...>
      wrote:
      >
      > --- In primenumbers@yahoogroups.com, "Mark Underwood"
      > <mark.underwood@s...> wrote:
      > >
      > ...
      > > x^2 - 2*y^2 = 100999 should (and does) have exactly one solution.
      > >
      > ...
      > > Mark
      > >
      >
      > Solutions of x^2 - 2*y^2 = 100999:
      >
      > x0=357, y0=115
      >
      > Xn+1 = 3 Xn + 4 Yn
      > Yn+1 = 2 Xn + 3 Yn
      >
      > so we can deduce:
      >
      > x1=1531, y1=1059
      > x2=8829, y2=6239
      > x3=51443, y3=36375
      > x4=299829, y4=212011
      > x5=1747531, y5=1235691
      > x6=10185357, y6=7202135
      > x7=59364611, y7=41977119
      > x8=346002309, y8=244660579
      > x9=2016649243, y9=1425986355
      > ....
      >
      > We can also negate x_n and/or y_n.
      >
      > You can find the solution using my Quadratic Diophantine Equation
      > Solver at:
      >
      > http://www.alpertron.com.ar/QUAD.HTM
      >
      > Best regards,
      >
      > Dario Alpern
      > Buenos Aires - Argentina
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