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.
Here are a few proofs: 1, 2 (which is excellent), and the somewhat bizarre 3.
Unfortunately, there are no cases where you have nontrivial arithmetic progressions of higher powers. This is a string of proofs. Carmichael himself covered this for n = 3 and 4, about a hundred years ago. But it wasn’t completed until Ribet wrote a paper on it in the 90s. His paper can be found here. The statement is equivalent to when we let $\alpha = 1$. Funny enough, he happens to have sent out a notice on scimath with a little humor, which can still be found here.
A quick Google search found this: http://www.math.ku.dk/~kiming/lecture_notes/2007-2008-elliptic_curves/4_squares_in_arithmetic_progression.pdf . It contains a sketch of an elementary proof at the end and cites Dickson’s History of the theory of numbers.
My favourite proof of this is Van der Poorten’s — it uses descent, as Fermat almost certainly would have.