question on trying to use quadratic residues to eliminate needless checks

Hello:

I am trying to determine if using quadratic residues will shorten down 
the time to check two numbers, a,b such that a^2 + b^2 = square (say s^2)

In regard to body cuboids, I looked at the side = 288807105787200

Using the divisors of side^2, we find that 1,492,627 integers or 
fractions exist, such that x^2 + 288807105787200^2 = s^2 where x is in Z 
or Q (actually n/2)

There are 1,262,992 integers with this property and 229,635 rationals 
(all have denominator 2) with this property.

I am looking for two numbers r,s from this pile of 1,492,627 values such 
that r^2 + s^2 = t^2 (t either integer or rational)

Two examples:

288807105787200^2 + 1944335775346650^2 = 1965668118387750^2

288807105787200^2 + (2560559305082625/2)^2 = (2624900404254975/2)^2

Naively running through this batch of 1,492,627 values requires 
1,113,966,934,251 checks to see if r^2 + s^2 is a square.

This is taxing, even for my 6 core Intel Xeon 3.5 Mhz system and 
requires several days to run.

Is there a way, using quadratic residues, to eliminate most of the pair 
checks and reduce the running time?

Randall

P.S. I have been running for about a day now and only about 1/4 the way 
done.