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I have been looking for integer solutions of certain Diophantine equations, one of the simplest examples being

$x^2=4+8y^2+13z^2$.

The ideal answer would be a way to parametrize all the integer equations. It is easy to find infinitely many solutions. For instance, by setting $y=z$, we can use the solutions of Pell's equation. Is there a more general technique to get other solutions? Note that the techniques used for the Legendre's equation do not apply since this equation is not homogeneous.

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    $\begingroup$ One can parametrize all solutions. I recommend the first and last paragraph of my response to a similar question: mathoverflow.net/questions/114707/… $\endgroup$
    – GH from MO
    Commented Apr 20, 2015 at 20:59
  • $\begingroup$ In General, the solution of this equation is determined by the solutions of the Pell equations. Approximately in this form. math.stackexchange.com/questions/74931/… $\endgroup$
    – individ
    Commented Apr 21, 2015 at 4:48
  • $\begingroup$ @GH, Thank you for your answer. Its good to know that it can be solved. I went also further considering the case (not important for my purposes) where the coefficients are $+1$ and $-1$, but not as further as you did. However, this case with coefficients $4, 8, 13$ appears to be more delicate. $\endgroup$
    – Mendes
    Commented Apr 21, 2015 at 7:18
  • $\begingroup$ Dear @GH, I have looked through your answer to mathoverflow.net/questions/114707/. However, this special case is not quite helpful for me! Could you please kindly post an answer describing an effective algorithm of determination of the orbits of the integral automorphism group of a quadratic form in the set of integer solutions in the general case, say on the example of the equation $x^2-8y^2-13z^2=4$? If necessary, I can learn $p$-adic numbers and adeles in order to understand your answer! $\endgroup$
    – Mendes
    Commented Apr 21, 2015 at 21:16
  • $\begingroup$ Your problem is a special case of Theorem 6.2 in Chapter 13 of Cassels: Rational quadratic forms. This theorem says that there is an algorithm to find the reduced representations (whose number is finite), and every other representation is an image of a reduced one by an automorph of $-x^2+8y^2+13z^2$. So it remains to find the group of automorphs, for which Sections 4,5,7,8 of Chapter 13 in the book might be helpful. $\endgroup$
    – GH from MO
    Commented Apr 21, 2015 at 21:51

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