A problem related to modern methods of prime factorization involve finding subsets of a random sequence of integers whose product is a perfect square. There is a threshold phenomena where the existence of such a subset transitions from being very unlikely to very likely once a size parameter of the sequence reaches a certain point. This threshold informs us about the expected runtime of the prime factorization algorithm. I will detail this connection, what theorems are known, and what unresolved questions I am pursuing.