The Impossible Puzzle

I'm leaving this one here not necessarily as a puzzle for you to solve, but as an insight in how freaky these math puzzles can get. This particular one is called ‘The Impossible Puzzle’. Solving it is in fact possible, but that title might hint at how difficult that is. Nevertheless, see how far you can get!

Sam and Priya are two very talented mathematicians.
Their friend Anna approaches them and says: “I have chosen two whole numbers, labeled A and B. Note that A is greater than 1 and B is greater than A. The sum of these numbers does not exceed 100.”

The others nod, so she continues: “In a moment, I will inform Sam of only the sum (A+B), and I will inform Priya of only the product (A×B). These announcements remain private!”
She does so, and the following conversation then takes place:

Priya says, “I don't know the values of A and B.”
Sam responds, “I already knew that.”
“In that case, I do know what their values are,” says Priya.
“Really?”, Sam ponders. “Then so do I.”

Now you too, can know what numbers A and B are.

Solution

That seems rather ridiculous. We're hardly being told anything at all. And yet, using some advanced math and clever reasoning, the answer can be deduced. Here's a tip: the puzzle takes the concept of this earlier one and increases it difficulty a trillion-fold.

As for explaining the solution... uhm... that goes behind the scope of this blog. Okay, honestly: it's well out of my league.

Luckily, the solution (A =  4  and B = 13, click to see) is explained here and also here.