Khinchin A.Y. Three Pearls of Number Theory

Translated by F. Bagemihl, H.Komm and W. Seidel. — New York: Dover Publications, 1998. — 62 p.Exact facsimile of the original edition, not reproduced with Optical Recognition Software. The author, one of the leading Russian mathematicians of the post-war period, attempts to present three important results in number theory in such a way as to promote interest in the subject by showing that elementary number theory is not yet a finished field and that highly interesting new results can still be obtained by ingenious methods. These three 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. Proofs and explanations of the answers are provided.