Each of X and Y is a positive integer which is not a perfect square that satisfy:

√X - √Y = √17

Is each of X and Y always divisible by 17?

Give reasons for your answer.

When shuffling a deck of cards using a riffle shuffle, one divides the deck in two and lets the two halves riffle down to the table, interleaving as they do so. Assume that a person using this shuffle will always divide a deck of 52 cards exactly evenly, and that the riffle will start equally often from the left as from the right.

The expert dealer that I am, when I perform a riffle shuffle the cards from the two halves always interleave perfectly, the cards alternating from the left and right halves of the deck.

How many times must I shuffle the deck before the probability of correctly guessing the next card down in the deck after seeing a card chosen randomly from some place in the deck will be less than 1.97%? (If the cards were perfectly random, the probability of correctly guessing the next card would be 1/51 = 1.96%)

Bonus: What if there were a 10% chance that, as each card falls during the riffle, the card will be covered by another card from the same half, instead of strictly alternating?

(*Assume that the person guessing knows the original order of the cards, the number of times the deck has been shuffled, and the probability of the cards interleaving perfectly.*)