## Re: primes and squares

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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
Message 1 of 6 , Apr 4, 2005
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:
>