# Additive Number Theory The Classical Bases

Springer Science & Business Media, Jun 25, 1996 - Mathematics - 342 pages
[Hilbert's] style has not the terseness of many of our modem authors in mathematics, which is based on the assumption that printer's labor and paper are costly but the reader's effort and time are not. H. Weyl [143] The purpose of this book is to describe the classical problems in additive number theory and to introduce the circle method and the sieve method, which are the basic analytical and combinatorial tools used to attack these problems. This book is intended for students who want to lel?Ill additive number theory, not for experts who already know it. For this reason, proofs include many "unnecessary" and "obvious" steps; this is by design. The archetypical theorem in additive number theory is due to Lagrange: Every nonnegative integer is the sum of four squares. In general, the set A of nonnegative integers is called an additive basis of order h if every nonnegative integer can be written as the sum of h not necessarily distinct elements of A. Lagrange 's theorem is the statement that the squares are a basis of order four. The set A is called a basis offinite order if A is a basis of order h for some positive integer h. Additive number theory is in large part the study of bases of finite order. The classical bases are the squares, cubes, and higher powers; the polygonal numbers; and the prime numbers. The classical questions associated with these bases are Waring's problem and the Goldbach conjecture.

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### Contents

 III 3 IV 4 V 5 VI 7 VII 12 VIII 17 IX 24 X 27
 LI 178 LII 186 LIII 191 LIV 195 LV 199 LVI 204 LVII 208 LIX 211

 XI 33 XII 34 XIII 37 XIV 38 XV 44 XVI 49 XVII 71 XVIII 72 XIX 75 XXI 77 XXII 86 XXIII 94 XXV 97 XXVII 99 XXVIII 102 XXIX 103 XXX 111 XXXI 118 XXXIII 121 XXXV 124 XXXVI 125 XXXVII 127 XXXVIII 129 XXXIX 133 XL 137 XLI 146 XLII 147 XLIV 151 XLV 153 XLVI 158 XLVII 167 XLVIII 173 XLIX 174 L 177
 LX 212 LXI 213 LXII 215 LXIII 220 LXIV 227 LXV 230 LXVII 231 LXVIII 238 LXIX 244 LXX 251 LXXI 259 LXXII 267 LXXIV 271 LXXV 272 LXXVI 275 LXXVII 279 LXXVIII 281 LXXIX 286 LXXX 292 LXXXI 297 LXXXII 298 LXXXIII 301 LXXXIV 303 LXXXV 308 LXXXVI 310 LXXXVII 314 LXXXVIII 317 LXXXIX 320 XC 323 XCI 327 XCIII 331 XCIV 341 Copyright