Given the Diophantine equation $$ r^2-ds^2 = x^2-dy^2 = q, $$ (where $q$ is a potentially unknown integer, and certainly need not be $1$), the two solutions $(r,s)$ and $(x,y)$ are called equivalent if and only if there exist integers $t,u$ such that $t^2-du^2=1$ and $(x,y)=(rt+dsu,ru+st)$ [Robertson etc.].

What relationships and/or transformations exist between non-equivalent solutions? Has anyone done serious research on this, e.g. what does it mean if $\gcd(s,y)>1$?

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    $\begingroup$ suggest you read my answers on Conway's topograph/river method. Meanwhile, C.L. Siegel initiated, for indefinite forms, a way to count the number of orbits of an entire genus representing a number; an orbit would be all your "equivalent" solutions. Siegel is intricate, the comparison is to a rational number to be calculated from p-adic information, similar to the Mass Formula en.wikipedia.org/wiki/… $\endgroup$ – Will Jagy Jan 7 '15 at 4:05
  • $\begingroup$ Anyway, the number of orbits is finite, generally quite small, grows more or less by the number of distinct prime factors of your $q.$ The different solutions come from different $\pm$ choices in en.wikipedia.org/wiki/Brahmagupta#Pell.27s_equation $\endgroup$ – Will Jagy Jan 7 '15 at 4:21
  • $\begingroup$ Write down the solution of the equation $$r^2-ds^2=x^2-dy^2$$ Then come to the binary quadratic form. And decisions will be recorded by the solutions of the Pell equation. Conditions for the existence of solutions and their formulas I wrote. $\endgroup$ – individ Jan 7 '15 at 4:59
  • $\begingroup$ There is an algorithm to solve the generalized Pell equation. You can find the solutions and then comment about the non-equivalent solutions using the structure of the solution. $\endgroup$ – MathGod Jan 7 '15 at 11:33
  • $\begingroup$ @MathGod: Yes, that's the Robertson I referenced in my original post. I was hoping not to reinvent the wheel, if someone's already done the analysis. $\endgroup$ – Kieren MacMillan Jan 8 '15 at 14:09

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