mathematics - Generalization of Sum and Product puzzle

I heard from a clever person that for the problem mentioned before: I don't know the two numbers... but now I do a solution exists even if we change 100 to infinity.

So the formulation would be next:

Two perfect logicians, Summer and Proctor, are told that integers x and y have been chosen such that $x>1$ and $y>1$. Summer is given the value x+y and Proctor is given the value x⋅y. They then have the following conversation.

Proctor: "I cannot determine the two numbers."
Summer: "I knew that."
Proctor: "Now I can determine them."
Summer: "So can I."

Given that the above statements are true, what are the two numbers?

I don't know whether it is true. Can you find the solution or prove that it doesn't exists? Can you find a general solution for any integer $N$ if we have limitation $x+y