The Impossible puzzle ("Now I know your product")

Let $x,y$ be integers greater than $1$, $P=xy$ and $S=x+y$.
Write $P=x_1\cdots x_n$, product of not necessarily distinct primes. If $n=2$, then necessarily $S=x_1+x_2$, so, if $S$ isn't the sum of two primes (this case), knowing $P$ tells nothing about $S$.
Then, we know that $n\ge 3$ and $S$ isn't the sum of two primes ($S$ isn't even, in particular, and then necessarily $P$ is even.).

So, necessarily $x$ is even and $y$ is odd. If we write $P=2^k p_1$, where $p_1$ is any prime, then necessarily $x=2^k$ and $y=p_1$, so, in this case $S$ would be known. Then, in this case, $P=2^k p_1\cdots p_m$, with $m>1$ and $p_i$ prime (since in this case knowing $P$ tells nothing about $S$).

So $S$ is odd and the set $\{xy:x+y=S,x,y>1\}$ contains one and only one number of the form $P=2^kp_1\cdots p_m$, with $p_i$ prime and $m>1$.

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In your link, the statement is a little different:

Sam (to Polly): "You can't know what x and y are."
Polly (to Sam): "That was true, but now I do."
Sam (to Polly): "Now I do too."

So, in this case we have $S$ odd, isn't the sum of two primes, and the set $\{xy:x+y=S,x,y>1\}$ contains one and only one number of the form $2^kp$, with $p$ odd prime, $k>1$, and $P=2^kp$, $x=2^k$, $y=p$. This excludes several possibilities and the rest is brute force (and we need an upper bound for $x$ and $y$).