Product and Sum (unsolved)

(Seen in a presentation by Rineke Verbrugge, in NVTI Theory day, 2007)
Of two unknown integers, each between 2 and 99 inclusive, a person P is told the product and a person S is told the sum. When asked whether they know the two numbers, the following dialog ensues:

P: “I don’t know them.”
S: “I knew that already.”
P: “Then I now know the two numbers.”
S: “Then I now know them, too.” 

What are the two numbers? Prove that your solution is unique.

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1000 bottles of wine and 10 rats

(Can be found in Leino’s puzzle page)
A mad scientist has 1000 bottles of whine, but one of them is poisoned. He also has 10 rats, for which the poison bottle will kill in any amount, within 10 days.
The scientist wants to have a party on the 11th day, and remove the poisoned bottle until there. How can he find the right bottle?