The Impossible Pair

Two whole numbers are chosen. Each is at least two, and their sum is at most one hundred. The numbers may be equal.

Sara is told their sum. Pavel is told their product. Both know the rules above, and both know that the other has been told their half.

Pavel says: “I do not know the two numbers.”

Sara says: “I knew you would not. I do not know them either.”

Pavel says: “Now I know them.”

Sara says: “Then so do I.”

What are the two numbers?