These quadratic residues are then examined very closely to see if they do indeed factorize into small primes as required. Finding this set is non-trivial, and is not performed by the general worker machines. For each ppr, we draw an arc between the nodes labelled by the two large-ish primes in the relation. We then go around looking for closed loops, aka cycles, in that network. That is, the large-ish prime appears as a square when the two corresponding quadratic residues are multiplied together.
Squares are good, remember.
We do this as many times as possible, to generate as many different combinations as possible, none of which contain any large-ish primes. We know that we need a very nearly square matrix to stand any credible chance of success. Gaussian elimination on matrices this size is also non-trivial. Anyway, if we find such a dependency, we have found our x and y as required.