Riddles, brain puzzles and mathematical problems

[BAndrew]

Here's more:

• Prove that all primes greater than 2 can be written as the difference of two squares.
  
  
• Decode this:
  31 51 02 51 81 90 91 02 91 10
  81 50 10 91 11 50 40 02 51 20
  50 51 41 02 80 50 21 51 51 11
  51 12 02 60 51 81 91 90 42 02
  50 50 41 80 10 81 40 50 41 50
  40 30 81 90 31 90 41 10 21 91
  
  
• Your sister and your best friend have been acting rather suspiciously recently. One evening you notice your sister secretly pass a piece of paper into your friend's pocket. You wait for the right opportunity and remove the note to discover the following:
  
  
  R1U0R0L4L6D5L7D1 L4U6R0R2U4 U6L0D8 R1D1R2.
  R1U0L7L4D1U8R2R1 L2L3L8D4D5L2D3!
  
  Confused, you record the secret message and return the piece of paper. The next day, while your sister is out, you look around the house and discover a scrap of paper in one of the waste paper bins. It has the following similar message written on it, but it seems to contain a clue:
  
  
  R2D1R1L8 = TEST
  
  Can you decode the original message and work out what your sister and best friend are up to?
  
  
• Prove that n must be odd for (1+1/2)(1+1/3)(1+1/4)...(1+1/n) to be an integer.
  
  
• Find the next number:
  2,3,5,7,11,13,?