You are hired by french NSA to break the RSA code used on the Pink Card. The easiest way to do that is to factor the public modulus and you have found the fastest algorithm to do that, except that you have to solve a subproblem that can be modeled in the following way.
Let be a set of prime numbers. If and are formed with elements of , then S*T will denote the quantity

We call relation a set of two primes p,q, where p and q are distinct elements of . You dispose of a collection of R relations and you are interested in finding sequences of these, such that

is a perfect square.

The way you look for these squares is the following. The ultimate goal is to count squares that appear in the process. Relations arrive one at a time. You maintain a collection of relations that do not contain any square subproduct. This is easy: at first, is empty. Then a relation arrives and begins to grow. Suppose a new relation arrives. If no square appears when adding to , then is added to the collection. Otherwise, a square is about to appear, we increase the number of squares, but we do not store this relation, hence keeps the desired property.
Let us consider an example. First arrives and we put it in ; then arrives and they are stored in . Now enters the relation . This relation could be used to form the square:

So we count 1 and do not store in . Now we consider that could make a square with , so we count 1 square more. Then is put into . Now could make the square . Eventually, we get 3 squares.