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There's a lot of interesting trivia related to this. For instance, the sequence of perfect squares {1,4,9,...} contains 3-term subsequences that are in arithmetic progression (e.g., {1,25,49}). However, it does not contain any 4-term such subsequences. And if you consider the sequence of nth powers, for n > 2, it does not contain any arithmetic progressions at all. This claim turns out to be very similar to Fermat's Last Theorem and can be established with similar techniques. (See https://math.berkeley.edu/~ribet/Articles/acta.pdf, with alpha = 1. Note that this is the same Ribet of Ribet's theorem, which established the link between modularity of elliptic curves and FLT, setting the stage for Wiles.)

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