Once in a while I run across cool or surprising theorems, like Midy's theorem which says that the repeating digits in the decimal expansion of (for instance) 1/17 = 0.0588235294117647... can be split into two halves and summed to give 05882352 + 94117647 = 99999999. Another is Szemererdi's_theorem which says that "any 'positive fraction' of the positive integers will contain arbitrarily long arithmetic progressions", that is, sequences whose terms are obtained by starting at some number a and adding some number r repeatedly.
Anybody have any other nifty theorems?
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