Hardy and Wright – Theory of Numbers

Abnormal Number  Subject Index  Euler’s Identity

Index for

An Introduction to THE THEORY OF NUMBERS

by

G.H. Hardy and E.M. Wright

Published

by

Oxford University Press, London

compiled

by Robert E. Kennedy

and Curtis Cooper

Central Missouri State University

Hardy and Wright’s The Theory of Numbers was published in 1938 and is now in its fifth edition (1979). The authors admitted that there were large gaps in their book and that the topics were presented with very little depth. But why did this book become such a classic? In our opinion, the preface to the first edition indicates the reason. There, the authors write that their own personal interests dictated the material to be included and chose topics that they considered “congenial”. Thus,as they stated in the preface, they could hardly have failed because “...the subject matter being so attractive that only extravagant incompetence could make it dull.”

So, what is the purpose of compiling an index for a classic volume that is probably one of the most respected number theory books of this century? Because it doesn’t have one!! It has always seemed, to us, that this had to be an oversight on the part of Hardy and Wright or their publishers. We believe that a good index for a mathematics book enhances the viability of it as a reference for research and study. Hopefully, neither of the authors would mind us constructing an index for their book.

This index is valid for both the 4th and 5th editions.

INDEX

Abnormal Number . . . . . . . . . . . . . . . 21 Additive Theory of Numbers . . . . . . . . . . . 273 Algebraic Number . . . . . . . . . . . . . 159, 178 Algebraic Irrational . . . . . . . . . . . . . . . 39 Algebraic Equation . . . . . . . . . . . . . . . 159 Algebraic Integer . . . . . . . . . . . . . . . . 178 Algebraic Field . . . . . . . . . . . . . . . . . 204 Algebraic Number . . . . . . . . . . . . . . . 204 Almost All . . . . . . . . . . . . . . . . . . 8, 122 Arithmetic of Quadratic Fields . . . . . . . . . . 225 Arithmetical Progression . . . . . . . . . . . . . 113 Arithmetical Functions . . . . . . . . . . . . . 232 Associate . . . . . . . . . . . . . 67, 181, 183, 305 Associate (mod m) . . . . . . . . . . . . . . . 89 Asymptotically Equivalent . . . . . . . . . . . . . 8 Average Order . . . . . . . . . . . . . . . 263, 272 Bachet’s Weights Problem . . . . . . . . . . . . 115 Bauer’s Identical Congruence . . . . . . . 98, 100, 102 Belongs to . . . . . . . . . . . . . . . . . . . 71 Bernoulli’s Numbers . . . . . . . . . . . . . . . 90 Bertrand’s Postulate . . . . . . . . . . . . . . 343 Big-Oh Notation . . . . . . . . . . . . . . . . . 7 Binary Decimal . . . . . . . . . . . . . . . . . 111 Binomial Coefficient . . . . . . . . . . . . . . . 63 Biquadrate . . . . . . . . . . . . . . . . . 317, 327 Bohr’s Proof . . . . . . . . . . . . . . . . . . 388 Boundary . . . . . . . . . . . . . . . . . . . 31 Bounded Quotients . . . . . . . . . . . . . . . 165 Cantor’s Ternary Set . . . . . . . . . . . . . . 124 Chinese Remainder Theorem (Theorem 121) …. 95 Circular Representation . . . . . . . . . . . . . 390 Class of Residues . . . . . . . . . . . . . . . . 49 Closed Region . . . . . . . . . . . . . . . . . 31 Closed Set . . . . . . . . . . . . . . . . . . . 121 Combinatorial Proof . . . . . . . . . . . . . . . 278 Complete System of Residues . . . . . . . . . 49, 220 Complete Set of Residues Prime to m . . . . . . . 52 Composite Integer . . . . . . . . . . . . . . . . 2 Congruence . . . . . . . . . . . . . . . . . . 49 Conjugate Partitions . . . . . . . . . . . . . . 274 Conjugate . . . . . . . . . . . . . . . . . . . 305 Continued Fraction . . . . . . . . . . . . . . . 127 Continued Fraction Algorithm . . . . . . . . . . 134 Convergent . . . . . . . . . . . . . . . 128, 151, 164 Convex Region . . . . . . . . . . . . . . . . . 31 Coprime . . . . . . . . . . . . . . . . . . . . 48 Decimal . . . . . . . . . . . . . . . . . . . . 107 Dedekind Section . . . . . . . . . . . . . . . . 377 Degree . . . . . . . . . . . . . . . . . . . . . 204 Dense in Itself . . . . . . . . . . . . . . . . . 121 Dense . . . . . . . . . . . . . . . . . . . . . 377 Derived Set . . . . . . . . . . . . . . . . 121, 377 Determinant . . . . . . . . . . . . . . . . . . 397 Digits (missing) . . . . . . . . . . . . . . . 120, 122 Diophantine Equation . . . . . . . . . . . . 190, 191 Dirichlet Series . . . . . . . . . . . . . 244, 248, 259 Dirichlet’s Theorem . . . . . . . . . . 13, 18, 93, 373 Dirichlet’s Argument . . . . . . . . . . . . 156, 176 Dirichlet’s Divisor Problem . . . . . . . . . . . . 272 Divisibility of Polynomials (mod m) . . . . . . . . 83 Divisibility Tests . . . . . . . . . . . . . . . . 114 Divisibility in k(i) . . . . . . . . . . . . . . . . 182 Divisibility (in an extension field) . . . . . . . . . 208 Divisible . . . . . . . . . . . . . . . . . . . . . 1 Divisible (with respect to Ideals) . . . . . . . . . 228 Divisor (in an extension field) . . . . . . . . . . 208 Durfee Square . . . . . . . . . . . . . . . . . 281 Enumerable Set . . . . . . . . . . . . . . . . . 121 Equivalent Points . . . . . . . . . . . . . . . . 35 Equivalent Numbers . . . . . . . . . . . . . . . 141 Estemann’s Proof . . . . . . . . . . . . . . . . 386 Euclid Number . . . . . . . . . . . . . . . . . 240 Euclid’s First Theorem . . . . . . . . . . . . . . 3 Euclid’s Second Theorem . . . . . . . . . . . 4, 13 Euclid’s Theorem . . . . . . . . . . . . . 14, 16, 18 Euclid’s Algorithm . . . . . . . . . . . 136, 179, 212 Euclidean Construction . . . . . . . . . . . 58, 159 Euclidean Number . . . . . . . . . . . . . . . 159 Euclidean Field . . . . . . . . . . . . . . . . . 212 Euclidean Quadratic Field . . . . . . . . . . . . 213 Euler’s Constant . . . . . . . . . . . . 39, 264, 351 Euler’s Function . . . . . . . . . . . . . . . . 52 Euler’s Identity . . . . . . . . . . . . . . . 284, 285

Euler’s Conjecture  Subject Index  Primitive Root

Euler’s Conjecture . . . . . . . . . . . . . . . 332 Euler-Maclaurin Sum Formula . . . . . . . . . . 90 Even Convergent . . . . . . . . . . . . . . . . 132 Excluded Interval . . . . . . . . . . . . . . . . 377 Farey Series . . . . . . . . . . . . . 23, 30, 36, 268 Farey Arc . . . . . . . . . . . . . . . . . . . 30 Farey Dissection . . . . . . . . . . . . . . . . 30 Farey Point . . . . . . . . . . . . . . . . . . 30 Fermat Number . . . . . . . . . . . . . . . . . 14 Fermat Prime . . . . . . . . . . . . . . . . 19, 58 Fermat’s Conjecture . . . . . . . . . . . . . 6, 14, 18 Fermat’s Theorem . . . . . . . . . 63, 71, 85, 86, 87 Fermat’s Last Theorem . . . . . . . 73, 190, 202, 231 Fermat’s Theorem in k(i) . . . . . . . . . . . . 219 Fermat’s Problem . . . . . . . . . . . . . . . . 332 Fermat-Euler Theorem . . . . . . . . . . . . . 63 Ferrier’s Prime . . . . . . . . . . . . . . . . . 22 Fibonacci Series . . . . . . . . . . . . . . 148, 153 Four-Square Problem . . . . . . . . . . . . 302, 315 Frequency (of a digit) . . . . . . . . . . . . . . 124 Fundamental Theorem of Arithmetic . . . . . . 3, 21,

179, 180, 185, 188, 211

Fundamental Point-Lattice . . . . . . . . . . . . 26 Fundamental Lattice . . . . . . . . . . . . . . 26 Fundamental Parallelogram . . . . . . . . . . . 34 Fundamental Theorem of Arithmetic . . . . . . . 246 Gauss’s Sum . . . . . . . . . . . . . . . . . . 54 Gauss’s Lemma . . . . . . . . . . . . . . . . . 74 Gaussian Integer . . . . . . . . . . . . 178, 182, 189 Generating Function . . . . . . . . . . . . . . 244 Goldbach’s Theorem(conjecture) . . . . . . . . 19, 22 Highest Common Divisor . . . . . . . . . 20, 48, 186 Highest Common Right-Hand Divisor . . . . . . . 307 Ideal . . . . . . . . . . . . . . . . . . . . . 227 Index . . . . . . . . . . . . . . . . . . . . . 71 Integer . . . . . . . . . . . . . . . . . . . . . 1 Integers of k(ρ) . . . . . . . . . . . . . . . . . 187 Integral Lattice . . . . . . . . . . . . . . . . . 26 Integral Polynomial . . . . . . . . . . . . . . . 82 Integral Quaternion . . . . . . . . . . . . . 304, 306 Interior Point . . . . . . . . . . . . . . . . . . 31 Irrational Number . . . . . . . . . . . . . 38, 112 Jacobi’s Theorem . . . . . . . . . . . . . . . . 282 Kloosterman’s Sum . . . . . . . . . . . . . . . 56 Kronecker’s Theorem . . . . . . . 375, 382, 384, 393 Lagrange’s Proof . . . . . . . . . . . . . . . . 87 Lagrange’s Theorem . . . . . . . . . . . . . . . 302 Lambert Series . . . . . . . . . . . . . . . . . 257 Lattice . . . . . . . . . . . . . . . . . . . . 26 Lattice Point . . . . . . . . . . . . . . . . . . 264 Least Common Multiple . . . . . . . . . . . . . 48 Least Residue . . . . . . . . . . . . . . . . . 49 Legendre’s Symbol . . . . . . . . . . . . . . 68, 80 Legendre’s Theorem . . . . . . . . . . . . . . . 320 Lettenmeyer’s Proof . . . . . . . . . . . . . . . 384 Leudesdorf’s Theorem . . . . . . . . . . . . . . 100 Limit Point . . . . . . . . . . . . . . . . . . 121 Linear Conguences . . . . . . . . . . . . . . 51, 94 Linearly Independent . . . . . . . . . . . . 379, 381 Liouville’s Theorem . . . . . . . . . . . . . . . 161 Little-Oh Notation . . . . . . . . . . . . . . . . 7 Logarithmic Function . . . . . . . . . . . . . . . 8 Lucas Series . . . . . . . . . . . . . . . . . . 148 Lucas’s Test . . . . . . . . . . . . . . 16, 223, 231 Maximum Period . . . . . . . . . . . . . . . . 114 Measure Zero . . . . . . . . . . . . . . . . . . 121 Mediant . . . . . . . . . . . . . . . . . . . . 23 Mersenne Prime . . . . . . . . . . . . . . 18, 240 Mersenne Number . . . . . . . . . 14, 80, 148, 224 Merten’s Theorem . . . . . . . . . . . . . . . 351 Mesh . . . . . . . . . . . . . . . . . . . . . 376 Method of Descent . . . . . . . . . . . . . 194, 300 Minimal Residue . . . . . . . . . . . . . . . . 73 Minkowski’s Theorem . . . . . . . . . . . . . . 32 Minkowski’s Theorem (Converse) . . . . . . . . . 407 Mobius Inversion Formula . . . . . . . . . . 236, 251 Mobius Function . . . . . . . . . . . . 234, 243, 360 Moduli . . . . . . . . . . . . . . . . . . . . 19 Multiplicative Function . . . . . . . . . . . 53, 235 Neighbourhood . . . . . . . . . . . . . . . . . 121 Nim . . . . . . . . . . . . . . . . . . . . . . 117 Non-homogenous Forms . . . . . . . . . . . . . 402 Non-Negative Integer . . . . . . . . . . . . . . . 1 Norm of an Integer . . . . . . . . . . . . . . . 182 Norm . . . . . . . . . . . . . . . . . . . . . 309 Normal Numbers . . . . . . . . . . . . . . . . 124 Normal Order . . . . . . . . . . . . . . . . . 356 Null Modulus . . . . . . . . . . . . . . . . . . 20 Null Set . . . . . . . . . . . . . . . . . . 122, 168 Number

Abnormal . . . . . . . . . . . . . . . . . 21 Algebraic . . . . . . . . . . . . . . . 159, 178 Algebraic . . . . . . . . . . . . . . . . . 204 Bernoulli’s . . . . . . . . . . . . . . . . . 90 Equivalent . . . . . . . . . . . . . . . . . 141 Euclid . . . . . . . . . . . . . . . . . . . 240 Euclidean . . . . . . . . . . . . . . . . . 159 Fermat . . . . . . . . . . . . . . . . . . 14 Irrational . . . . . . . . . . . . . . . 38, 112 Mersenne . . . . . . . . . . . 14, 80, 148, 224 Normal . . . . . . . . . . . . . . . . . . 124 Perfect . . . . . . . . . . . . . . . . . . 239 Quadratfrei . . . . . . . . . . . . . . . . 269 Round . . . . . . . . . . . . . . . . . . . 358 Transcendental . . . . . . 159, 160, 170, 173, 177 Triangular . . . . . . . . . . . . . . . . . 284 Odd Convergent . . . . . . . . . . . . . . . . 132 Open Region . . . . . . . . . . . . . . . . . . 31 Order of Magnitude . . . . . . . . . . . . . . 7, 260 Order of a mod m . . . . . . . . . . . . . . . 71 Order of Approximation . . . . . . . . . . . . . 158 Partition . . . . . . . . . . . . . . . . . . . . 273 Pell’s Equation . . . . . . . . . . . . . . . . . 217 Perfect Set . . . . . . . . . . . . . . . . . . . 121 Perfect Number . . . . . . . . . . . . . . . . . 239 Periodic Continued Fractions . . . . . . . . . . . 143 Point-Lattice . . . . . . . . . . . . . . . . . . 26 Positive Integer . . . . . . . . . . . . . . . . . . 1 Positive Definite . . . . . . . . . . . . . . . . 397 Primality Tests . . . . . . . . . . . . . . . . . 78 Prime Integer . . . . . . . . . . . . . . . . . . 2 Prime Number Theorem (Theorem 6) . . . . . . 9, 374 Prime in k(1) . . . . . . . . . . . . . . . . . . 181 Prime in k(i) . . . . . . . . . . . . . . . . 183, 219 Prime (in an extension field) . . . . . . . . . . . 208 Prime (with respect to Ideals) . . . . . . . . . . 228 Prime Quaternions . . . . . . . . . . . . . . . 309 Prime Pairs . . . . . . . . . . . . . . . . . . 371 Primitive Root of Unity . . . . . . . . . . . . . 55 Primitive Root . . . . . . . . . . . . . . . 71, 115

Primitive Polynomial  Subject Index  Zeta Function

Primitive Polynomial . . . . . . . . . . . . . . 205 Principal Ideal . . . . . . . . . . . . . . . . . 229 Principle Right-Ideal . . . . . . . . . . . . . . 307 Prouhet and Tarry’s Problem . . . . . . . . . . . 328 Pure Recurring Decimal . . . . . . . . . . . . . 110 Pythagoras’ Theorem . . . . . . . . . . . . . 39, 42 Quadratfrei . . . . . . . . . . . . . . . . . . 16 Quadratfrei Scale . . . . . . . . . . . . . . . . 112 Quadratfrei Number . . . . . . . . . . . . . . . 269 Quadratic Residue . . . . . . . . . . . . . . . 67 Quadratic Non-Residue . . . . . . . . . . . . . 68 Quadratic Surd . . . . . . . . . . . . . . . 144, 146 Quadratic Field . . . . . . . . . . . . . . . 204, 206 Quadratic Form . . . . . . . . . . . . . . . . 396 Quaternion . . . . . . . . . . . . . . . . . 303, 316 Ramanujan’s Sum . . . . . . . . . . . . . . 55, 237 Ramanujan’s Continued Fraction . . . . . . . . . 295 Rational Integer . . . . . . . . . . . . . . . 1, 178 Rational Approximation . . . . . . . . . . . 163, 166 Real Euclidean Field . . . . . . . . . . . . . . 213 Reciprocity Law . . . . . . . . . . . . . . . . 76 Recurring Decimal . . . . . . . . . . . . . . . 109 Reflected Ray Problem . . . . . . . . . . . . . 378 Regular Polygon . . . . . . . . . . . . . . . . 57 Residue . . . . . . . . . . . . . . . . . . . 49, 87 Riemann Zeta Function . . . . . . . . . . . . . 245 Right-Ideal . . . . . . . . . . . . . . . . . . . 307 Rogers-Ramanujan Identities . . . . . . . . . 290, 296 Root of f(x) (mod m) . . . . . . . . . . . . . . 82 Round Number . . . . . . . . . . . . . . . . . 358 Scale (Base) . . . . . . . . . . . . . . . . . . 111 Selberg’s Theorem . . . . . . . . . . . . . . . 360 Self-Conjugate Partition . . . . . . . . . . . . . 278 Set of Points . . . . . . . . . . . . . . . . . . 121 Sieve of Eratosthenes . . . . . . . . . . . . . . . 3 Simple Continued Fraction . . . . . . . . 131, 132, 138 Simple Field . . . . . . . . . . . . . . . . . . 212 Simply Normal . . . . . . . . . . . . . . . . . 124 Simultaneous Approximation . . . . . . . . . . . 169 Square Lattice . . . . . . . . . . . . . . . . . 229 Standard Form . . . . . . . . . . . . . . . . . . 2 Star Region . . . . . . . . . . . . . . . . . . 410 Tchebotaref’s Theorem . . . . . . . . . . . 405, 413 Tchebychef’s Theorem . . . . . . . . . . . 9, 345, 373 Terminating Decimal . . . . . . . . . . . . . . 109 Theorem

Euclid’s First . . . . . . . . . . . . . . . . . 3 Euclid’s Second . . . . . . . . . . . . . . 4, 13 Euclid’s Second . . . . . . . . . . . . . . 4, 13 Euclid’s . . . . . . . . . . . . . . . 14, 16, 18 Fermat’s . . . . . . . . . . . . 63, 71, 85, 86, 87 Fermat’s Last . . . . . . . . . 73, 190, 202, 231 Fermat’s in k(i) . . . . . . . . . . . . . . . 219 Fermat-Euler . . . . . . . . . . . . . . . . 63 Goldbach’s (Conjecture) . . . . . . . . . . 19, 22 Jacobi’s . . . . . . . . . . . . . . . . . . 282 Kronecker’s . . . . . . . . . 375, 382, 384, 393 Lagendre’s . . . . . . . . . . . . . . . . . 320 Lagrange’s . . . . . . . . . . . . . . . . . 302 Leudesdorf’s . . . . . . . . . . . . . . . . 100 Liouville’s . . . . . . . . . . . . . . . . . 161 Mertin’s . . . . . . . . . . . . . . . . . . 351 Minkowski’s . . . . . . . . . . . . . . 32, 407 Prime Number (Theorem 6) . . . . . . . . 9, 374 Pythagoras’ . . . . . . . . . . . . . . . 39, 42 Selberg’s . . . . . . . . . . . . . . . . . . 360 Tchebotaref’s . . . . . . . . . . . . . 405, 413 Tchebychef’s . . . . . . . . . . . . . 9, 345, 373 Von Staudt’s . . . . . . . . . . . . . . . . 90 Wilson’s . . . . . . . . . . . . . . 68,81,86,87 Wolstenhome’s . . . . . . . . . . . . 88, 93,100 Three-Square Problem . . . . . . . . . . . . . . 316 Transcendental Number …. 159, 160, 170, 173, 177 Triangular Number . . . . . . . . . . . . . . . 284 Uniform Distribution . . . . . . . . . . . . . . 390 Unimodular Transformation . . . . . . . . . . . 28 Unities of k(1) . . . . . . . . . . . . . . . . . 181 Unity in k(i) . . . . . . . . . . . . . . . . . . 182 Unity (in an extension field) . . . . . . . . . . . 208 Unity . . . . . . . . . . . . . . . . . . . . . 305 Vector . . . . . . . . . . . . . . . . . . . . . 376 Visible Point . . . . . . . . . . . . . . . . 29, 409 Von Staudt’s Theorem . . . . . . . . . . . . . . 90 Vulgar Fraction . . . . . . . . . . . . . . . . . 23 Waring’s Problem . . . . . . . . . 297, 317, 325, 335 Wilson’s Theorem . . . . . . . . . . . 68, 81, 86, 87 Wolstenholme’s Theorem . . . . . . . . . 88, 93, 100 Zeta Function . . . . . . . . . . . . . . . . . 245