Three Pearls of Number Theory

These 3 puzzles involve the proof of a basic law governing the world of numbers known to be correct in all tested cases — the problem is to prove that the law is always 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. 

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Faced with the unrelieved tedium of a hospital stay, a Russian soldier recuperating from his World War II injuries begged one of his former instructors for something to occupy his mind. The professor replied with "Three Pearls of Number Theory", a trio of mathematical problems as absorbing today as they were more than 50 years ago.

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These 3 puzzles require proof of a basic law governing the world of numbers. Features van der Waerden's theorem, the Landau-Schnirelmann hypothesis and Mann's theorem, and a solution to Waring's problem. Solutions included.