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