An Index for Hardy & Wright's The Theory of Numbers

By Chris Caldwell

An Index for
An Introduction to THE THEORY OF NUMBERS
by G.H. Hardy and E.M. Wright
(published by the Oxford University Press, London)

This index 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.

[ ABCDEFGHIJKLMNOPQRSTUVWXYZ]

• 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, 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
• Mertens' 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, 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

(Placed on the web with the permission of the authors.)