# Problem 359: Hilbert's New Hotel
An infinite number of people (numbered 1, 2, 3, etc.) are lined up to
get a room at Hilbert's newest infinite hotel. The hotel contains an
infinite number of floors (numbered 1, 2, 3, etc.), and each floor
contains an infinite number of rooms (numbered 1, 2, 3, etc.). Initially
the hotel is empty. Hilbert declares a rule on how the nth person is
assigned a room: person n gets the first vacant room in the lowest
numbered floor satisfying either of the following: the floor is empty
the floor is not empty, and if the latest person taking a room in that
floor is person m, then m + n is a perfect square Person 1 gets room 1
in floor 1 since floor 1 is empty. Person 2 does not get room 2 in floor
1 since 1 + 2 = 3 is not a perfect square. Person 2 instead gets room 1
in floor 2 since floor 2 is empty. Person 3 gets room 2 in floor 1 since
1 + 3 = 4 is a perfect square. Eventually, every person in the line gets
a room in the hotel. Define P(f, r) to be n if person n occupies room r
in floor f, and 0 if no person occupies the room. Here are a few
examples: P(1, 1) = 1 P(1, 2) = 3 P(2, 1) = 2 P(10, 20) = 440 P(25, 75)
= 4863 P(99, 100) = 19454 Find the sum of all P(f, r) for all positive f
and r such that f × r = 71328803586048 and give the last 8 digits as
your answer.