An introduction to the theory of numbers:
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Oxford [u.a.]
Oxford Univ. Press
2008
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| Ausgabe: | sixth edition, new edition material |
| Schriftenreihe: | Oxford mathematics
|
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| Beschreibung: | XXI, 621 Seiten |
| ISBN: | 9780199219865 9780199219858 |
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| 100 | 1 | |a Hardy, Godfrey H. |d 1877-1947 |e Verfasser |0 (DE-588)118720376 |4 aut | |
| 245 | 1 | 0 | |a An introduction to the theory of numbers |c G. H. Hardy and E. M. Wright |
| 250 | |a sixth edition, new edition material | ||
| 264 | 1 | |a Oxford [u.a.] |b Oxford Univ. Press |c 2008 | |
| 300 | |a XXI, 621 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Oxford mathematics | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 4 | |a Number theory | |
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| 689 | 0 | 0 | |a Zahlentheorie |0 (DE-588)4067277-3 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Wright, Edward M. |d 1906-2005 |e Verfasser |0 (DE-588)174091729 |4 aut | |
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CONTENTS
Foreword by Andrew Wiles v
Preface to the sixth edition
vii
Preface to the fifth edition
viii
Preface to the first edition
ix
Remarks on notation
xi
I. THE SERIES OF PRIMES
( 1 ) 1
1.1. Divisibility of integers
1
1.2.
Prime numbers
2
1.3.
Statement of the fundamental theorem of arithmetic
3
1.4.
The sequence of primes
4
1.5.
Some questions concerning primes
6
1.6.
Some notations
7
1.7.
The logarithmic function
9
1.8.
Statement of the prime number theorem
10
II. THE SERIES OF PRIMES
(2) 14
2.1.
First proof of Euclid's second theorem
14
2.2.
Further deductions from Euclid's argument
14
2.3.
Primes in certain arithmetical progressions
15
2.4.
Second proof of Euclid's theorem
17
2.5.
Fermat's and Mersenne's numbers
18
2.6.
Third proof of Euclid's theorem
20
2.7.
Further results on formulae for primes
21
2.8.
Unsolved problems concerning primes
23
2.9.
Moduli of integers
23
2.10.
Proof of the fundamental theorem of arithmetic
25
2.11.
Another proof of the fundamental theorem
26
III. FAREY SERIES AND A THEOREM OF MINKOWSKI
28
3.1.
The definition and simplest properties of a Farey series
28
3.2.
The equivalence of the two characteristic properties
29
3.3.
First proof of Theorems
28
and
29 30
3.4.
Second proof of the theorems
31
3.5.
The integral lattice
32
3.6.
Some simple properties of the fundamental lattice
33
3.7.
ThirdproofofTheorems28and29
35
3.8.
The Farey dissection of the continuum
36
3.9.
A theorem of Minkowski
37
3.10.
Proof of Minkowski's theorem
39
3.11.
Developments of Theorem
37 40
xiv CONTENTS
IV.
IRRATIONAL
NUMBERS
45
4.1.
Some generalities
45
4.2.
Numbers known to be irrational
46
4.3.
The theorem of Pythagoras and its generalizations
47
4.4.
The use of the fundamental theorem in the proofs of Theorems
43-45 49
4.5.
A historical digression
50
4.6.
Geometrical proof of the irrationality of
д/5
52
4.7.
Some more irrational numbers
53
V. CONGRUENCES AND RESIDUES
57
5.1.
Highest common divisor and least common multiple
57
5.2.
Congruences and classes of residues
58
5.3.
Elementary properties of congruences
60
5.4.
Linear congruences
60
5.5.
Euler's function
ф(т)
63
5.6.
Applications of Theorems
59
and
61
to trigonometrical sums
65
5.7.
A general principle
70
5.8.
Construction of the regular polygon of
17
sides
71
VI. FERMAT'S THEOREM AND ITS CONSEQUENCES
78
6.1.
Fermat's theorem
78
6.2.
Some properties of binomial coefficients
79
6.3.
A second proof of Theorem
72 81
6.4.
Proof of Theorem
22 82
6.5.
Quadratic residues
83
6.6.
Special cases of Theorem
79:
Wilson's theorem
85
6.7.
Elementary properties of quadratic residues and non-residues
87
6.8.
The order of a (mod m)
88
6.9.
The converse of Fermat's theorem
89
6.10.
Divisibility of IP-1
- 1
by p1
91
6.11.
Gauss's lemma and the quadratic character of
2 92
6.12.
The law of reciprocity
95
6.13.
Proof of the law of reciprocity
97
6.14.
Tests for primality
98
6.15.
Factors of Merserme numbers; a theorem of
Euler
100
VII.
GENERAL PROPERTIES OF CONGRUENCES
103
7.1.
Roots of congruences
103
7.2.
Integral polynomials and identical congruences
103
7.3.
Divisibility of polynomials (mod m)
105
7.4.
Roots of congruences to a prime modulus
106
7.5.
Some applications of the general theorems
108
CONTENTS xv
7.6.
Lagrange's proof of Fermat's and Wilson's theorems
110
7.7.
The residue of
{2
(p
- 1)}!
Ill
7.8.
A theorem of Wolstenholme
112
7.9.
The theorem of
von
Staudt
115
7.10.
Proof of
von Staudťs
theorem
116
VIII.
CONGRUENCES TO COMPOSITE MODULI
120
8.1.
Linear congruences
120
8.2.
Congruences of higher degree
122
8.3.
Congruences to a prime-power modulus
123
8.4.
Examples
125
8.5.
Bauer's identical congruence
126
8.6.
Bauer's congruence: the case p=2
129
8.7.
A theorem of Leudesdorf
130
8.8.
Further consequences of Bauer's theorem
132
8.9.
The residues of 2P~
'
and
(
ρ
- 1 )!
to modulus p2
135
IX. THE REPRESENTATION OF NUMBERS BY DECIMALS
138
9.1.
The decimal associated with a given number
138
9.2.
Terminating and recurring decimals
141
9.3.
Representation of numbers in other scales
144
9.4.
Irrationals defined by decimals
145
9.5.
Tests for divisibility
146
9.6.
Decimals with the maximum period
147
9.7.
Bachet's problem of the weights
149
9.8.
The game of
Nim
151
9.9.
Integers with missing digits
154
9.10.
Sets of measure zero
155
9.11.
Decimals with missing digits
157
9.12.
Normal numbers
158
9.13.
Proof that almost all numbers are normal
160
X. CONTINUED FRACTIONS
165
10.1.
Finite continued fractions
165
10.2.
Convergents
to a continued fraction
166
10.3.
Continued fractions with positive quotients
168
10.4.
Simple continued fractions
169
10.5.
The representation of an irreducible ratiooal fraction by a simple
continued fraction
170
10.6.
The continued fraction algorithm and Euclid's algorithm
172
10.7.
The difference between the fraction and its
convergents
175
10.8.
Infinite simple continued fractions
177
xvi CONTENTS
10.9.
The representation of an irrational number by an infinite
continued fraction
178
10.10.
A lemma
180
10.1
1
.
Equivalent numbers
181
10.12.
Periodic continued fractions
184
10.13.
Some special quadratic surds
187
10.14.
The series of Fibonacci and Lucas
190
10.15.
Approximation by
convergents
194
XI. APPROXIMATION OF IRRATIONALS BY RATIONALS
198
11.1.
Statement of the problem
198
11.2.
Generalities concerning the problem
199
11.3.
An argument of Dirichlet
201
11.4.
Orders of approximation
202
11.5.
Algebraic and transcendental numbers
203
11.6.
The existence of transcendental numbers
205
11.7.
Liouville's theorem and the construction of transcendental numbers
206
11.8.
The measure of the closest approximations to an arbitrary irrational
208
11.9.
Another theorem concerning the
convergents
to a continued fraction
210
11.10.
Continued fractions with bounded quotients
212
11.11.
Further theorems concerning approximation
216
11.12.
Simultaneous approximation
217
11.13.
The transcendence of
e
218
11.14.
The transcendence of
π
223
XII.
THE FUNDAMENTAL THEOREM OF ARITHMETIC IN
¿(I),*«, AND
¿(ρ)
229
12.1.
Algebraic numbers and integers
229
12.2.
The rational integers, the Gaussian integers, and the integers of k(p)
230
12.3.
Euclid's algorithm
231
12.4.
Application of Euclid's algorithm to the fundamental theorem in k(
1 ) 232
12.5.
Historical remarks on Euclid's algorithm and the fundamental theorem
234
12.6.
Properties of the Gaussian integers
235
12.7.
Primes in /fc(i)
236
12.8.
The fundamental theorem of arithmetic in k(i)
238
12.9.
The integers of k(p)
241
XIII.
SOME DIOPHANTINE EQUATIONS
245
13.1.
Fermat's last theorem
245
13.2.
The equation x2
+
y2
=
z2
245
13.3.
The equation x4
+
y4
=
z4
247
13.4.
The equation
x3+JÍ3
=z3
248
CONTENTS
xvii
13.5.
The equationXі +y3=3z3
253
13.6.
The expression of a rational as a sum of rational cubes
254
13.7.
The equation x3+y3+z3=t3
257
XIV.
QUADRATIC FIELDS
(1) 264
14.1.
Algebraic fields
264
14.2.
Algebraic numbers and integers; primitive polynomials
265
14.3.
The general quadratic field
кЏт)
267
14.4.
Unities and primes
268
14.5.
The unities of k(Jl)
270
14.6.
Fields in which the fundamental theorem is false
273
14.7.
Complex Euclidean fields
274
14.8.
Real Euclidean fields
276
14.9.
Real Euclidean fields (continued)
279
XV. QUADRATIC FIELDS
(2) 283
15.1.
The primes of it (i)
283
15.2.
Fermat's theorem in jfc(i)
285
15.3.
The primes of k(p)
286
15.4.
The primes of
кЏ1)
and k(J5)
287
15.5.
Lucas's test for the primality of the Mersenne number
Мап+з
290
15.6.
Generai
remarks on the arithmetic of quadratic fields
293
15.7.
Ideals in a quadratic field
295
15.8.
Other fields
299
XVI.
THE ARITHMETICAL FUNCTIONS
ф{п),
ß(n),d(n),a(n),r(n) 302
16.1.
The function
ф(п)
302
16.2.
A further proof of Theorem
63 303
16.3.
The
Möbius
function
, 304
16.4.
The
Möbius
inversion formula
305
16.5.
Further inversion formulae
307
16.6.
Evaluation of Ramanujan's sum
308
16.7.
The functions d(n) and ak(n)
310
16.8.
Perfect numbers
311
16.9.
The function r(n)
313
16.10.
Proof of the formula for An)
315
XVII.
GENERATING FUNCTIONS OF ARITHMETICAL FUNCTIONS
318
17.1.
The generation of arithmetical functions by means of Dirichlet series
318
17.2.
The
zeta
function
320
17.3.
The behaviour of £(j) wheni
—► 1 321
Î7.4.
Multiplication of Dirichlet series
323
xviii CONTENTS
17.5.
The generating functions of some special arithmetical functions
326
17.6.
The analytical interpretation of the
Möbius
formula
328
17.7.
The function A(n)
331
17.8.
Further examples of generating functions
334
17.9.
The generating function of r(n)
337
17.10.
Generating functions of other types
338
XVIII.
THE ORDER OF MAGNITUDE OF ARITHMETICAL FUNCTIONS
342
18.1.
The order of
Ą
n)
342
18.2.
The average order of d(ri)
347
18.3.
The order of
σ
(я)
350
18.4.
The order of
ф(л)
352
18.5.
The average order of
ф(гі)
353
18.6.
The number of squarefree numbers
355
18.7.
The order of An)
356
XIX.
PARTITIONS
361
19.1.
The general problem of additive arithmetic
361
19.2.
Partitions of numbers
361
19.3.
The generating function of p(n)
362
19.4.
Other generating functions
365
19.5.
Two theorems of
Euler
366
19.6.
Further algebraical identities
369
19.7.
Another formula for F(x)
371
19.8.
A theorem of Jacobi
372
19.9.
Special cases of Jacobi's identity
375
19.10.
Applications of Theorem
353 378
19.11.
Elementary proof of Theorem
358 379
19.12.
Congruence properties ofp(n) %
380
19.13.
The Rogers-Ramanujan identities
383
19.14.
ProofofTheorems362and363
386
19.15.
Ramanujan's continued fraction
389
XX. THE REPRESENTATION OF A NUMBER BY TWO OR FOUR
SQUARES
393
20.1.
Waring's problem: the numbers g(k) and G(k)
393
20.2.
Squares
395
20.3.
Second proof of Theorem
366 395
20.4.
Third and fourth proofs of Theorem
366 397
20.5.
The four-square theorem
399
20.6.
Quaternions
401
20.7.
Preliminary theorems about integral quaternions
403
CONTENTS xix
20.8.
The highest common right-hand
divisor
of two quaternions
405
20.9.
Prime quaternions and the proof of Theorem
370 407
20.10.
The values of g(2) and G(2)
409
20.11.
Lemmas for the third proof of Theorem
369 410
20.12.
Third proof of Theorem
369:
the number of representations
411
20.13.
Representations by a larger number of squares
415
XXI.
REPRESENTATION BY CUBES AND HIGHER POWERS
419
21.1. Biquadrates 419
21.2.
Cubes: the existence of G(3) and g(3)
420
21.3.
A bound for g(3)
422
21.4.
Higher powers
424
21.5.
A lower bound for g(k)
425
21.6.
Lower bounds for G(k)
426
21.7.
Sums affected with signs: the number v{k)
431
21.8.
Upper bounds for v(k)
433
21.9.
The problem of Prouhet and Tarry: the number P(k,j)
435
21.10.
Evaluation of P(k,j) for particular
к
mà j
437
21.11.
Further problems of Diophantine analysis
440
XXII.
THE SERIES OF PRIMES
(3) 451
22.1.
The functions
ů(x)
and
ψ (χ)
451
22.2.
Proof that
ů
(χ)
and
ψ (χ)
are of order*
453
22.3.
Bertrand's postulate and a 'formula' for primes
455
22.4.
Proof of Theorems
7
and
9 458
22.5.
Two formal transformations
460
22.6.
An important sum
461
22.7.
The sum
Σρ~'
and the product
Π(1
-
p~l)
464
22.8.
Mertens's theorem
466
22.9.
Proof of Theorems
323
and
328 469
22.10.
The number of prime factors of
η
471
22.11.
The normal order of
ω
(л)
and
Ω
(л)
473
22.12.
A note on round numbers
476
22.13.
The normal order of din)
477
22.14.
Selberg's theorem
478
22.15.
The functions R(x) and
Υ(ξ)
481
22.16.
Completion of the proof of Theorems
434, 6,
and
8 486
22.17.
Proof of Theorem
335 489
22.18.
Products of A prime factors
490
22.19.
Primes in an interval
494
22.20.
A conjecture about the distribution of prime pairs p,p + 2
495
xx CONTENTS
XXIII. KRONECKER'
S
THEOREM 501
23.1.
Kronecker's theorem in one dimension
501
23.2.
Proofs of the one-dimensional theorem
502
23.3.
The problem of the reflected ray
505
23.4.
Statement of the general theorem
508
23.5.
The two forms of the theorem
510
23.6.
An illustration
512
23.7.
Lettenmeyer's proof of the theorem
512
23.8.
Estermann's proof of the theorem
514
23.9.
Bohr's proof of the theorem
517
23.10.
Uniform distribution
520
XXIV.
GEOMETRY OF NUMBERS
523
24.1.
Introduction and restatement of the fundamental theorem
523
24.2.
Simple applications
524
24.3.
Arithmetical proof of Theorem
448 527
24.4.
Best possible inequalities
529
24.5.
The best possible inequality for
ξ2 + η2
530
24.6.
The best possible inequality for
\ξη\
532
24.7.
A theorem concerning non-homogeneous forms
534
24.8.
Arithmetical proof of Theorem
455 536
24.9.
Tchebotarefs theorem
537
24.10.
A converse of Minkowski's Theorem
446 540
XXV.
ELLIPTIC CURVES
549
25.1.
The congruent number problem
549
25.2.
The addition law on an elliptic curve
550
25.3.
Other equations that define elliptic curves
556
25.4.
Points of finite order
559
25.5.
The group of rational points
564
25.6.
The group of points modulo p.
■ 573
25.7.
Integer points on elliptic curves
574
25.8.
The £-series of an elliptic curve
578
25.9.
Points of finite order and modular curves
582
25.10.
Elliptic curves and Fermat's last theorem
586
APPENDIX
593
1.
Another formula for
p„
593
2.
A generalization of Theorem
22 593
3.
Unsolved problems concerning primes
594
CONTENTS xxi
A LIST OF BOOKS
597
INDEX OF SPECIAL SYMBOLS AND WORDS
601
INDEX OF NAMES
605
GENERALINDEX
611 |
| adam_txt |
CONTENTS
Foreword by Andrew Wiles v
Preface to the sixth edition
vii
Preface to the fifth edition
viii
Preface to the first edition
ix
Remarks on notation
xi
I. THE SERIES OF PRIMES
( 1 ) 1
1.1. Divisibility of integers
1
1.2.
Prime numbers
2
1.3.
Statement of the fundamental theorem of arithmetic
3
1.4.
The sequence of primes
4
1.5.
Some questions concerning primes
6
1.6.
Some notations
7
1.7.
The logarithmic function
9
1.8.
Statement of the prime number theorem
10
II. THE SERIES OF PRIMES
(2) 14
2.1.
First proof of Euclid's second theorem
14
2.2.
Further deductions from Euclid's argument
14
2.3.
Primes in certain arithmetical progressions
15
2.4.
Second proof of Euclid's theorem
17
2.5.
Fermat's and Mersenne's numbers
18
2.6.
Third proof of Euclid's theorem
20
2.7.
Further results on formulae for primes
21
2.8.
Unsolved problems concerning primes
23
2.9.
Moduli of integers
23
2.10.
Proof of the fundamental theorem of arithmetic
25
2.11.
Another proof of the fundamental theorem
26
III. FAREY SERIES AND A THEOREM OF MINKOWSKI
28
3.1.
The definition and simplest properties of a Farey series
28
3.2.
The equivalence of the two characteristic properties
29
3.3.
First proof of Theorems
28
and
29 30
3.4.
Second proof of the theorems
31
3.5.
The integral lattice
32
3.6.
Some simple properties of the fundamental lattice
33
3.7.
ThirdproofofTheorems28and29
35
3.8.
The Farey dissection of the continuum
36
3.9.
A theorem of Minkowski
37
3.10.
Proof of Minkowski's theorem
39
3.11.
Developments of Theorem
37 40
xiv CONTENTS
IV.
IRRATIONAL
NUMBERS
45
4.1.
Some generalities
45
4.2.
Numbers known to be irrational
46
4.3.
The theorem of Pythagoras and its generalizations
47
4.4.
The use of the fundamental theorem in the proofs of Theorems
43-45 49
4.5.
A historical digression
50
4.6.
Geometrical proof of the irrationality of
д/5
52
4.7.
Some more irrational numbers
53
V. CONGRUENCES AND RESIDUES
57
5.1.
Highest common divisor and least common multiple
57
5.2.
Congruences and classes of residues
58
5.3.
Elementary properties of congruences
60
5.4.
Linear congruences
60
5.5.
Euler's function
ф(т)
63
5.6.
Applications of Theorems
59
and
61
to trigonometrical sums
65
5.7.
A general principle
70
5.8.
Construction of the regular polygon of
17
sides
71
VI. FERMAT'S THEOREM AND ITS CONSEQUENCES
78
6.1.
Fermat's theorem
78
6.2.
Some properties of binomial coefficients
79
6.3.
A second proof of Theorem
72 81
6.4.
Proof of Theorem
22 82
6.5.
Quadratic residues
83
6.6.
Special cases of Theorem
79:
Wilson's theorem
85
6.7.
Elementary properties of quadratic residues and non-residues
87
6.8.
The order of a (mod m)
88
6.9.
The converse of Fermat's theorem
89
6.10.
Divisibility of IP-1
- 1
by p1
91
6.11.
Gauss's lemma and the quadratic character of
2 92
6.12.
The law of reciprocity
95
6.13.
Proof of the law of reciprocity
97
6.14.
Tests for primality
98
6.15.
Factors of Merserme numbers; a theorem of
Euler
100
VII.
GENERAL PROPERTIES OF CONGRUENCES
103
7.1.
Roots of congruences
103
7.2.
Integral polynomials and identical congruences
103
7.3.
Divisibility of polynomials (mod m)
105
7.4.
Roots of congruences to a prime modulus
106
7.5.
Some applications of the general theorems
108
CONTENTS xv
7.6.
Lagrange's proof of Fermat's and Wilson's theorems
110
7.7.
The residue of
{2
(p
- 1)}!
Ill
7.8.
A theorem of Wolstenholme
112
7.9.
The theorem of
von
Staudt
115
7.10.
Proof of
von Staudťs
theorem
116
VIII.
CONGRUENCES TO COMPOSITE MODULI
120
8.1.
Linear congruences
120
8.2.
Congruences of higher degree
122
8.3.
Congruences to a prime-power modulus
123
8.4.
Examples
125
8.5.
Bauer's identical congruence
126
8.6.
Bauer's congruence: the case p=2
129
8.7.
A theorem of Leudesdorf
130
8.8.
Further consequences of Bauer's theorem
132
8.9.
The residues of 2P~
'
and
(
ρ
- 1 )!
to modulus p2
135
IX. THE REPRESENTATION OF NUMBERS BY DECIMALS
138
9.1.
The decimal associated with a given number
138
9.2.
Terminating and recurring decimals
141
9.3.
Representation of numbers in other scales
144
9.4.
Irrationals defined by decimals
145
9.5.
Tests for divisibility
146
9.6.
Decimals with the maximum period
147
9.7.
Bachet's problem of the weights
149
9.8.
The game of
Nim
151
9.9.
Integers with missing digits
154
9.10.
Sets of measure zero
155
9.11.
Decimals with missing digits
157
9.12.
Normal numbers
158
9.13.
Proof that almost all numbers are normal
160
X. CONTINUED FRACTIONS
165
10.1.
Finite continued fractions
165
10.2.
Convergents
to a continued fraction
166
10.3.
Continued fractions with positive quotients
168
10.4.
Simple continued fractions
169
10.5.
The representation of an irreducible ratiooal fraction by a simple
continued fraction
170
10.6.
The continued fraction algorithm and Euclid's algorithm
172
10.7.
The difference between the fraction and its
convergents
175
10.8.
Infinite simple continued fractions
177
xvi CONTENTS
10.9.
The representation of an irrational number by an infinite
continued fraction
178
10.10.
A lemma
180
10.1
1
.
Equivalent numbers
181
10.12.
Periodic continued fractions
184
10.13.
Some special quadratic surds
187
10.14.
The series of Fibonacci and Lucas
190
10.15.
Approximation by
convergents
194
XI. APPROXIMATION OF IRRATIONALS BY RATIONALS
198
11.1.
Statement of the problem
198
11.2.
Generalities concerning the problem
199
11.3.
An argument of Dirichlet
201
11.4.
Orders of approximation
202
11.5.
Algebraic and transcendental numbers
203
11.6.
The existence of transcendental numbers
205
11.7.
Liouville's theorem and the construction of transcendental numbers
206
11.8.
The measure of the closest approximations to an arbitrary irrational
208
11.9.
Another theorem concerning the
convergents
to a continued fraction
210
11.10.
Continued fractions with bounded quotients
212
11.11.
Further theorems concerning approximation
216
11.12.
Simultaneous approximation
217
11.13.
The transcendence of
e
218
11.14.
The transcendence of
π
223
XII.
THE FUNDAMENTAL THEOREM OF ARITHMETIC IN
¿(I),*«, AND
¿(ρ)
229
12.1.
Algebraic numbers and integers
229
12.2.
The rational integers, the Gaussian integers, and the integers of k(p)
230
12.3.
Euclid's algorithm
231
12.4.
Application of Euclid's algorithm to the fundamental theorem in k(
1 ) 232
12.5.
Historical remarks on Euclid's algorithm and the fundamental theorem
234
12.6.
Properties of the Gaussian integers
235
12.7.
Primes in /fc(i)
236
12.8.
The fundamental theorem of arithmetic in k(i)
238
12.9.
The integers of k(p)
241
XIII.
SOME DIOPHANTINE EQUATIONS
245
13.1.
Fermat's last theorem
245
13.2.
The equation x2
+
y2
=
z2
245
13.3.
The equation x4
+
y4
=
z4
247
13.4.
The equation
x3+JÍ3
=z3
248
CONTENTS
xvii
13.5.
The equationXі +y3=3z3
253
13.6.
The expression of a rational as a sum of rational cubes
254
13.7.
The equation x3+y3+z3=t3
257
XIV.
QUADRATIC FIELDS
(1) 264
14.1.
Algebraic fields
264
14.2.
Algebraic numbers and integers; primitive polynomials
265
14.3.
The general quadratic field
кЏт)
267
14.4.
Unities and primes
268
14.5.
The unities of k(Jl)
270
14.6.
Fields in which the fundamental theorem is false
273
14.7.
Complex Euclidean fields
274
14.8.
Real Euclidean fields
276
14.9.
Real Euclidean fields (continued)
279
XV. QUADRATIC FIELDS
(2) 283
15.1.
The primes of it (i)
283
15.2.
Fermat's theorem in jfc(i)
285
15.3.
The primes of k(p)
286
15.4.
The primes of
кЏ1)
and k(J5)
287
15.5.
Lucas's test for the primality of the Mersenne number
Мап+з
290
15.6.
Generai
remarks on the arithmetic of quadratic fields
293
15.7.
Ideals in a quadratic field
295
15.8.
Other fields
299
XVI.
THE ARITHMETICAL FUNCTIONS
ф{п),
ß(n),d(n),a(n),r(n) 302
16.1.
The function
ф(п)
302
16.2.
A further proof of Theorem
63 303
16.3.
The
Möbius
function
, 304
16.4.
The
Möbius
inversion formula
305
16.5.
Further inversion formulae
307
16.6.
Evaluation of Ramanujan's sum
308
16.7.
The functions d(n) and ak(n)
310
16.8.
Perfect numbers
311
16.9.
The function r(n)
313
16.10.
Proof of the formula for An)
315
XVII.
GENERATING FUNCTIONS OF ARITHMETICAL FUNCTIONS
318
17.1.
The generation of arithmetical functions by means of Dirichlet series
318
17.2.
The
zeta
function
320
17.3.
The behaviour of £(j) wheni
—► 1 321
Î7.4.
Multiplication of Dirichlet series
323
xviii CONTENTS
17.5.
The generating functions of some special arithmetical functions
326
17.6.
The analytical interpretation of the
Möbius
formula
328
17.7.
The function A(n)
331
17.8.
Further examples of generating functions
334
17.9.
The generating function of r(n)
337
17.10.
Generating functions of other types
338
XVIII.
THE ORDER OF MAGNITUDE OF ARITHMETICAL FUNCTIONS
342
18.1.
The order of
Ą
n)
342
18.2.
The average order of d(ri)
347
18.3.
The order of
σ
(я)
350
18.4.
The order of
ф(л)
352
18.5.
The average order of
ф(гі)
353
18.6.
The number of squarefree numbers
355
18.7.
The order of An)
356
XIX.
PARTITIONS
361
19.1.
The general problem of additive arithmetic
361
19.2.
Partitions of numbers
361
19.3.
The generating function of p(n)
362
19.4.
Other generating functions
365
19.5.
Two theorems of
Euler
366
19.6.
Further algebraical identities
369
19.7.
Another formula for F(x)
371
19.8.
A theorem of Jacobi
372
19.9.
Special cases of Jacobi's identity
375
19.10.
Applications of Theorem
353 378
19.11.
Elementary proof of Theorem
358 379
19.12.
Congruence properties ofp(n) %
380
19.13.
The Rogers-Ramanujan identities
383
19.14.
ProofofTheorems362and363
386
19.15.
Ramanujan's continued fraction
389
XX. THE REPRESENTATION OF A NUMBER BY TWO OR FOUR
SQUARES
393
20.1.
Waring's problem: the numbers g(k) and G(k)
393
20.2.
Squares
395
20.3.
Second proof of Theorem
366 395
20.4.
Third and fourth proofs of Theorem
366 397
20.5.
The four-square theorem
399
20.6.
Quaternions
401
20.7.
Preliminary theorems about integral quaternions
403
CONTENTS xix
20.8.
The highest common right-hand
divisor
of two quaternions
405
20.9.
Prime quaternions and the proof of Theorem
370 407
20.10.
The values of g(2) and G(2)
409
20.11.
Lemmas for the third proof of Theorem
369 410
20.12.
Third proof of Theorem
369:
the number of representations
411
20.13.
Representations by a larger number of squares
415
XXI.
REPRESENTATION BY CUBES AND HIGHER POWERS
419
21.1. Biquadrates 419
21.2.
Cubes: the existence of G(3) and g(3)
420
21.3.
A bound for g(3)
422
21.4.
Higher powers
424
21.5.
A lower bound for g(k)
425
21.6.
Lower bounds for G(k)
426
21.7.
Sums affected with signs: the number v{k)
431
21.8.
Upper bounds for v(k)
433
21.9.
The problem of Prouhet and Tarry: the number P(k,j)
435
21.10.
Evaluation of P(k,j) for particular
к
mà j
437
21.11.
Further problems of Diophantine analysis
440
XXII.
THE SERIES OF PRIMES
(3) 451
22.1.
The functions
ů(x)
and
ψ (χ)
451
22.2.
Proof that
ů
(χ)
and
ψ (χ)
are of order*
453
22.3.
Bertrand's postulate and a 'formula' for primes
455
22.4.
Proof of Theorems
7
and
9 458
22.5.
Two formal transformations
460
22.6.
An important sum
461
22.7.
The sum
Σρ~'
and the product
Π(1
-
p~l)
464
22.8.
Mertens's theorem
466
22.9.
Proof of Theorems
323
and
328 469
22.10.
The number of prime factors of
η
471
22.11.
The normal order of
ω
(л)
and
Ω
(л)
473
22.12.
A note on round numbers
476
22.13.
The normal order of din)
477
22.14.
Selberg's theorem
478
22.15.
The functions R(x) and
Υ(ξ)
481
22.16.
Completion of the proof of Theorems
434, 6,
and
8 486
22.17.
Proof of Theorem
335 489
22.18.
Products of A prime factors
490
22.19.
Primes in an interval
494
22.20.
A conjecture about the distribution of prime pairs p,p + 2
495
xx CONTENTS
XXIII. KRONECKER'
S
THEOREM 501
23.1.
Kronecker's theorem in one dimension
501
23.2.
Proofs of the one-dimensional theorem
502
23.3.
The problem of the reflected ray
505
23.4.
Statement of the general theorem
508
23.5.
The two forms of the theorem
510
23.6.
An illustration
512
23.7.
Lettenmeyer's proof of the theorem
512
23.8.
Estermann's proof of the theorem
514
23.9.
Bohr's proof of the theorem
517
23.10.
Uniform distribution
520
XXIV.
GEOMETRY OF NUMBERS
523
24.1.
Introduction and restatement of the fundamental theorem
523
24.2.
Simple applications
524
24.3.
Arithmetical proof of Theorem
448 527
24.4.
Best possible inequalities
529
24.5.
The best possible inequality for
ξ2 + η2
530
24.6.
The best possible inequality for
\ξη\
532
24.7.
A theorem concerning non-homogeneous forms
534
24.8.
Arithmetical proof of Theorem
455 536
24.9.
Tchebotarefs theorem
537
24.10.
A converse of Minkowski's Theorem
446 540
XXV.
ELLIPTIC CURVES
549
25.1.
The congruent number problem
549
25.2.
The addition law on an elliptic curve
550
25.3.
Other equations that define elliptic curves
556
25.4.
Points of finite order
559
25.5.
The group of rational points
564
25.6.
The group of points modulo p.
■ 573
25.7.
Integer points on elliptic curves
574
25.8.
The £-series of an elliptic curve
578
25.9.
Points of finite order and modular curves
582
25.10.
Elliptic curves and Fermat's last theorem
586
APPENDIX
593
1.
Another formula for
p„
593
2.
A generalization of Theorem
22 593
3.
Unsolved problems concerning primes
594
CONTENTS xxi
A LIST OF BOOKS
597
INDEX OF SPECIAL SYMBOLS AND WORDS
601
INDEX OF NAMES
605
GENERALINDEX
611 |
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| author | Hardy, Godfrey H. 1877-1947 Wright, Edward M. 1906-2005 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T20:10:57Z |
| indexdate | 2025-01-16T15:01:06Z |
| institution | BVB |
| isbn | 9780199219865 9780199219858 |
| language | English |
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| physical | XXI, 621 Seiten |
| publishDate | 2008 |
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| spelling | Hardy, Godfrey H. 1877-1947 Verfasser (DE-588)118720376 aut An introduction to the theory of numbers G. H. Hardy and E. M. Wright sixth edition, new edition material Oxford [u.a.] Oxford Univ. Press 2008 XXI, 621 Seiten txt rdacontent n rdamedia nc rdacarrier Oxford mathematics Hier auch später erschienene, unveränderte Nachdrucke Number theory Zahlentheorie (DE-588)4067277-3 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Zahlentheorie (DE-588)4067277-3 s DE-604 Wright, Edward M. 1906-2005 Verfasser (DE-588)174091729 aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016394313&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hardy, Godfrey H. 1877-1947 Wright, Edward M. 1906-2005 An introduction to the theory of numbers Number theory Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4151278-9 |
| title | An introduction to the theory of numbers |
| title_auth | An introduction to the theory of numbers |
| title_exact_search | An introduction to the theory of numbers |
| title_exact_search_txtP | An introduction to the theory of numbers |
| title_full | An introduction to the theory of numbers G. H. Hardy and E. M. Wright |
| title_fullStr | An introduction to the theory of numbers G. H. Hardy and E. M. Wright |
| title_full_unstemmed | An introduction to the theory of numbers G. H. Hardy and E. M. Wright |
| title_short | An introduction to the theory of numbers |
| title_sort | an introduction to the theory of numbers |
| topic | Number theory Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Number theory Zahlentheorie Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016394313&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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