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
I've book marked your very cool site BTW!
--- In firstname.lastname@example.org
, "Dario Alpern" <alpertron@h...>
> --- In email@example.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:
> Best regards,
> Dario Alpern
> Buenos Aires - Argentina