These three puzzles involve the proof of a basic law governing the world of numbers known to be correct in all <i>tested</i> cases -- the problem is to prove that the law is <i>always</i> correct. Includes van der Waerden's theorem on arithmetic progressions, the Landau-Schnirelmann hypothesis and Mann's theorem, and a solution to Waring's problem. Proofs and explanations of the answers included.
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