This homework is due Thursday, November 13.

1. Verify the identities

      x2 - 1 == (x-1)(x-2) (mod 3)
      x4 - 1 == (x-1)(x-2)(x-3)(x-4) (mod 5)
   

2. Let p be a prime number. Prove that

      xp-1 - 1 == (x-1)(x-2) ... (x-(p-2))(x-(p-1) (mod p)
   

   Hint: You know something about the roots of the polynomial xp-1 -1 (mod p).

3. Read this fictional treatment of Hilbert's Hotel: www.c3.lanl.gov/mega-math/workbk/infinity/inhotel.html

4. Optional, but I'd really love to see it: rewrite the ending of the previous story so that the narrators scheme for housing the people on the infinitely many infinite busses succeeds (as it will). Then have the fire caused in some interesting way by the failure (discussed in class) of any scheme that claims to accomodate all the committees of guests.

5. In the on line demonstration of this mathematica program at http://demonstrations.wolfram.com/TheHilbertHotel/ you can see enough information to write a closed form expression for the room number for guest S_i,j. (The source code might help too.)