Is there any integer solution in-terms of $R,S$ for the equation $Rx^2+Sy^2=1$ , .
For example $(\frac{1}{\sqrt {2R}},\frac{1}{\sqrt {2S}})$ is a solution but not integer solution .
Is there any integer solution tuple for the equation in terms of R and S?
If not , is there any simple efficient algorithm to get integer solution ?
If $R=1$ and $S$ is a negative integer, this is Pell's equation. Algorithms exist to find solutions. I don't know how simple or efficient these algorithms are, though.
If you explore that route, you might find some information on the more general case where $R$ is a positive integer and $S$ is a negative integer.