## Three Pearls of Number TheoryThese 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 included. |

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appear arbi arbitrary natural number arbitrary number arithmetic progression assert assume average density basis of order belong ber of solutions brevity canonical extension chapter classes coefficients combination completes the proof construction definition Dover elementary arithmetic entire sequence equation 14 equation 26 equation 9 exceed the number form 9 form an arithmetic FOURIER SERIES fundamental lemma Gottingen Hence Hilbert's theorem hypotheses induction integers interval 2/V1/B k classes left half Let us denote linear equations mathematicians mathematics neighboring segments number n number n(k number of solutions number of systems NUMBER THEORY numbers is divided obtained PARTIAL DIFFERENTIAL EQUATIONS polynomial f(x positive density problem progression of segments proposition prove the fundamental quation quence result satisfy equation satisfy the inequalities Schnirelmann segment l,n segments of length sequel sequence of natural sequence of numbers simple solutions of equation subsegments summands theorem of Lagrange tions ural numbers values Waring's Waring's problem