trying to parameterize solutions for Pythagorean ratios and Diophantine m-tuples

[American Citizen <website.reader3@gmail.com>]

Subject: Trying to parameterize solutions for a product of two 
Pythagorean ratios + 1 made a square

I am trying to find solutions for equation (1) for certain rational 
values of a and b, where both are
Pythagorean ratios

(1)    a * b + 1 = s^2

A typical solution is a = 5/12, b = 63/16 and s = 13/8

Given a Pythagorean ratio, say a, I want to find a parametric or linear 
algorithm for quickly finding all
solutions of b which will work. Brute force computer work is not efficient.

We know that a Pythagorean ratio, parameterized by two integers, m,n; is 
r = (m+n)*(m-n)/(2*m*n) so both a,b
in (1) have this general form.

If I let b = (p+q)*(p-q)/(2*p*q) in (1) and solve, the equation becomes

(2)     mnpq (nq - mq + np + mp) (nq + mq - np + mp) = s^2  for 
m,n,p,q,s in Z

For the ratio 5/12 (m,n = 3,2) we obtain

(3)    30*q*p^3 + 144*q^2*p^2 - 30*q^3*p = r^2 for p,q,r in Z

This is almost an homogeneous quadratic equation in p,q made a square, 
but the product of the powers = 3
for each binomial term on the left.

Is there a parametric solution for (2) given m,n or (3) ?

For [2,1] <= [p,q] <= [1000,999] we find 20 solutions for [m,n] = [3,2].

    first   second ratio     m,n    p,q             S
    -----  --------------   -----  ----------    -------
     5/12  -4/3             [3,2]  [3,-1]          2/3
     5/12  -12/5            [3,2]  [5,-1]           0    (trivial)
     5/12   5/12            [3,2]  [3,2]          13/12  (trivial)
     5/12   63/16           [3,2]  [8,1]          13/8
     5/12  -21/20           [3,2]  [5,-2]          3/4
     5/12   99/20           [3,2]  [10,1]          7/4
     5/12   7812/125        [3,2]  [125,1]        26/5
     5/12   352/135         [3,2]  [27,5]         13/9
     5/12   1196/147        [3,2]  [49,3]         44/21
     5/12  -551/240         [3,2]  [24,-5]         5/24
     5/12   41760/289       [3,2]  [289,1]       133/17
     5/12  -1088/735        [3,2]  [49,-15]       13/21
     5/12   6956/3267       [3,2]  [121,27]      136/99
     5/12  -13617/7744      [3,2]  [121,-32]      91/176
     5/12   6409/24000      [3,2]  [125,96]      253/240
     5/12  -67137/73984     [3,2]  [289,-128]    429/544
     5/12  -339360/142129   [3,2]  [841,-169]     27/377
     5/12  -499675/212268   [3,2]  [722,-147]    221/1596
     5/12   288480/303601   [3,2]  [841,361]     651/551
     5/12   43092/320045    [3,2]  [605,529]     260/253

NOTE: My computer had to run through 499,500 trials to find these 20 
solutions, very inefficient.

Question and my goal:

   Can GP-Pari quickly find solutions to (3) or (2) for one given ratio 
from [m,n]

- Randall