TITLE: Squares in arithmetic progression
QUESTION [30 upvotes]: It is easy to find 3 squares (of integers) in arithmetic progression. For example, $1^2,5^2,7^2$.
I've been told Fermat proved that there are no progressions of length 4 in the squares. Do you know of a proof of this result? 
(Additionally, are there similar results for cubes, 4th powers, etc? If so, what would be a good reference for this type of material?)

Edit, March 30, 2012: The following question in MO is related and may be useful to people interested in the question I posted here.

REPLY [6 votes]: My favourite proof of this is Van der Poorten's — it uses descent, as Fermat almost certainly would have.