TITLE: Solving equations in a subset of rational numbers
QUESTION [6 upvotes]: Let $S$ be a set of all positive rational numbers $x$ such that $2x^2 - 1$ is a square, excluding $x=1$.
I am interested in computing as many as possible solutions in $S$ to either the following equations:
(1) $\qquad p^2 - 1 = (q^2 - 1)\cdot r^2$
(2) ...removed...
(3) $\qquad p^2 + q^2 = 1 + r^2$
What would a reasonable computational approach for finding solutions?
For the equation (1), I know one solution: $(p,q,r)=(\tfrac{373}{23}, \tfrac{85}{41}, \tfrac{205}{23})$ -- would it help to find more solutions? 
EDIT: Equation (2) was not exactly the one I'm interested in. So I removed it.

REPLY [5 votes]: Your question is equivalent to looking for rational points on certain hypersurfaces in affine 3-space. 
The equation $2x^2 - 1 = y^2$ describes a plane conic with a rational point $(x,y) = (1,1)$. This means you can parameterize its solutions by passing a line through $(1,1)$ and finding the other point of intersection:
$$
 x = \frac{t^2 - 2t + 2}{t^2 - 2} \qquad y = \frac{-t^2 + 4t - 2}{t^2 - 2}.
$$
Now you can substitute this expression for $x$ in for $p$, $q$, and $r$ (using a different variable for $t$ each time) to arrive at an equation for a hypersurface. For example, using the variables $a,b,c$, your equation (2) becomes
$$
 (a^2 - 2a + 2)(b^2 - 2b + 2)(c^2 - 2) = (c^2 - 2c + 2)(a^2 - 2)(b^2 - 2).
$$
At this point, I think I've transformed your question, but I don't immediately see how to produce rational points on these hypersurfaces.