Goldbach's conjecture
Let us call the split of a number "A" into two terms "A=B+C" a 2-split. There is no need for any advanced knowledge like Goldbach's conjecture or the fact that for the product "B·C" of such a 2-split to be unique (i.e. there are no other two numbers that also when multiplied yield the same result). But with Goldbach's conjecture, along with the fact that P would immediately know X and Y if their product were a semiprime, it can be deduced that the sum x+y cannot be even, since every even number can be written as the sum of two prime numbers. The product of those two numbers would then be a semiprime.