Softcover ISBN:  9781470463243 
Product Code:  CHEL/163.S 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $62.10 
eBook ISBN:  9781470454593 
Product Code:  CHEL/163.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $58.50 
Softcover ISBN:  9781470463243 
eBook: ISBN:  9781470454593 
Product Code:  CHEL/163.S.B 
List Price:  $134.00 $101.50 
MAA Member Price:  $120.60 $91.35 
AMS Member Price:  $120.60 $91.35 
Softcover ISBN:  9781470463243 
Product Code:  CHEL/163.S 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $62.10 
eBook ISBN:  9781470454593 
Product Code:  CHEL/163.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $58.50 
Softcover ISBN:  9781470463243 
eBook ISBN:  9781470454593 
Product Code:  CHEL/163.S.B 
List Price:  $134.00 $101.50 
MAA Member Price:  $120.60 $91.35 
AMS Member Price:  $120.60 $91.35 

Book DetailsAMS Chelsea PublishingVolume: 163; 1964; 309 ppMSC: Primary 11
A special feature of Nagell's wellknown text is the rather extensive treatment of Diophantine equations of second and higher degree. A large number of nonroutine problems are given.

Table of Contents

Front Cover

PREFACE

CONTENTS

CHAPTER I: DIVISIBILITY

1. Divisors.

2. Remainders.

3. Primes.

4. The fundamental theorem.

5. Least common multiple and greatest common divisor.

6. Moduls, rings and fields.

7. Euclid's algorithm.

8. Relatively prime numbers. Euler's φfunction.

9. Arithmetical functions.

10. Diophantine equations of the first degree.

11. Lattice points and point lattices.

12. Irrational numbers.

13. Irrationality of the numbers e and :rr.

Exercises

CHAPTER II: ON THE DISTRIBUTION OF PRIMES

14. Some lemmata.

15. General remarks. The sieve of Eratosthenes.

16. The function π(x).

17. Some elementary results on the distribution of primes.

18. Other problems and results concerning primes.

CHAPTER III: THEORY OF CONGRUENCES

19. Definitions and fundamental properties.

20. Residue classes and residue systems.

21. Fermat's theorem and its generalization by Euler.

22. Algebraic congruences and functional congruences.

23. Linear congruences.

24. Algebraic congruences to a prime modulus.

25. Prime divisors of integral polynomials.

26. Algebraic congruences to a composite modulus.

27. Algebraic congruences to a primepower modulus.

28. Numerical examples of solution of algebraic congruences.

29. Divisibility of integral polynomials with regard to a primemodulus.

30. Wilson's theorem and its generalization.

31. Exponent of an integer modulo n.

32. Moduli having primitive roots.

33. The index calculus

34. Power residues. Binomial congruences.

35. Polynomials representing integers.

36. Thue's remainder theorem and its generalization by Scholz.

Exercises

CHAPTER IV: THEORY OF QUADRATIC RESIDUES

37. The general quadratic congruence.

38. Euler's criterion and Legendre's symbol.

39. On the solvability of the congruences x2 ≡ ± 2 (mod p ).

40. Gauss's lemma.

41. The quadratic reciprocity law.

42. Jacobi's symbol and the generalization of the reciprocity law.

43. The prime divisors of quadratic polynomials.

44. Primes in special arithmetical progressions.

CHAPTER V: ARITHMETICAL PROPERTIES OF THE ROOTS OF UNITY

45. The roots of unity.

46. The cyclotomic polynomial.

47. Irreducibility of the cyclotomic polynomial.

48. The prime divisors of the cyclotomic polynomial.

49. A theorem of Bauer on the prime divisors of certain polynomials.

50. On the primes of the form ny  I.

51. Some trigonometrical products.

52. A polynomial identity of Gauss.

53. The Gaussian sums.

Exercises

CHAPTER VI: DIOPHANTINE EQUATIONS OF THE SECOND DEGREE

54. The representation of integers as sums of integral squares.

55. Bachet's theorem.

56. The Diophantine equation x2 Dy2= I.

57. The Diophantine equation x2 Dy2 = 1.

58. The Diophantine equation u2 D v2 = C

59. Lattice points on conics.

60. Rational points in the plane and on conics.

61. The Diophantine equation ax2 + by2 + cz2 = 0.

CHAPTER VII: DIOPHANTINE EQUATIONS OF HIGHER DEGREE

62. Some Diophantine equations of the fourth degree with three unknowns.

63. The Diophantine equation 2x4 y4= z2.

64. The quadratic fields K (√1), K (√ 2) and K (√3).

65. The Diophantine equation ξ3 + n3 + ζ3 = 0 and analogous equations.

66. Diophantine equations of the third degree with an infinityof solutions.

67. The Diophantine equation x7 + y7 + z7 = 0.

68, Fermat's last theorem,

69. Rational points on plane algebraic curves. Mordell's theorem.

70. Lattice points on plane algebraic curves. Theorems of Thue and Siegel.

Exercises

CHAPTER VIII: THE PRIME NUMBER THEOREM

71. Lemmata on the order of magnitude of some finite sums.

72. Lemmata on the Mohius function and some related functions.

73. Further lemmata. Proof of Selherg's formula.

74. An elementary proof of the prime number theorem.

Exercises

Table

The fundamental solutions of the equations x2 D y2 = ± 1 for D ⫹ 103.

NAME INDEX

SUBJECT INDEX

Back Cover


Reviews

This is a very readable introduction to number theory, with particular emphasis on diophantine equations, and requires only a school knowledge of mathematics. The exposition is admirably clear. More advanced or recent work is cited as background, where relevant ... [T]here are welcome novelties: Gauss's own evaluation of Gauss's sums, which is still perhaps the most elegant, is reproduced apparently for the first time. There are 180 examples, many of considerable interest, some of these being little known.
Mathematical Reviews


RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Reviews
 Requests
A special feature of Nagell's wellknown text is the rather extensive treatment of Diophantine equations of second and higher degree. A large number of nonroutine problems are given.

Front Cover

PREFACE

CONTENTS

CHAPTER I: DIVISIBILITY

1. Divisors.

2. Remainders.

3. Primes.

4. The fundamental theorem.

5. Least common multiple and greatest common divisor.

6. Moduls, rings and fields.

7. Euclid's algorithm.

8. Relatively prime numbers. Euler's φfunction.

9. Arithmetical functions.

10. Diophantine equations of the first degree.

11. Lattice points and point lattices.

12. Irrational numbers.

13. Irrationality of the numbers e and :rr.

Exercises

CHAPTER II: ON THE DISTRIBUTION OF PRIMES

14. Some lemmata.

15. General remarks. The sieve of Eratosthenes.

16. The function π(x).

17. Some elementary results on the distribution of primes.

18. Other problems and results concerning primes.

CHAPTER III: THEORY OF CONGRUENCES

19. Definitions and fundamental properties.

20. Residue classes and residue systems.

21. Fermat's theorem and its generalization by Euler.

22. Algebraic congruences and functional congruences.

23. Linear congruences.

24. Algebraic congruences to a prime modulus.

25. Prime divisors of integral polynomials.

26. Algebraic congruences to a composite modulus.

27. Algebraic congruences to a primepower modulus.

28. Numerical examples of solution of algebraic congruences.

29. Divisibility of integral polynomials with regard to a primemodulus.

30. Wilson's theorem and its generalization.

31. Exponent of an integer modulo n.

32. Moduli having primitive roots.

33. The index calculus

34. Power residues. Binomial congruences.

35. Polynomials representing integers.

36. Thue's remainder theorem and its generalization by Scholz.

Exercises

CHAPTER IV: THEORY OF QUADRATIC RESIDUES

37. The general quadratic congruence.

38. Euler's criterion and Legendre's symbol.

39. On the solvability of the congruences x2 ≡ ± 2 (mod p ).

40. Gauss's lemma.

41. The quadratic reciprocity law.

42. Jacobi's symbol and the generalization of the reciprocity law.

43. The prime divisors of quadratic polynomials.

44. Primes in special arithmetical progressions.

CHAPTER V: ARITHMETICAL PROPERTIES OF THE ROOTS OF UNITY

45. The roots of unity.

46. The cyclotomic polynomial.

47. Irreducibility of the cyclotomic polynomial.

48. The prime divisors of the cyclotomic polynomial.

49. A theorem of Bauer on the prime divisors of certain polynomials.

50. On the primes of the form ny  I.

51. Some trigonometrical products.

52. A polynomial identity of Gauss.

53. The Gaussian sums.

Exercises

CHAPTER VI: DIOPHANTINE EQUATIONS OF THE SECOND DEGREE

54. The representation of integers as sums of integral squares.

55. Bachet's theorem.

56. The Diophantine equation x2 Dy2= I.

57. The Diophantine equation x2 Dy2 = 1.

58. The Diophantine equation u2 D v2 = C

59. Lattice points on conics.

60. Rational points in the plane and on conics.

61. The Diophantine equation ax2 + by2 + cz2 = 0.

CHAPTER VII: DIOPHANTINE EQUATIONS OF HIGHER DEGREE

62. Some Diophantine equations of the fourth degree with three unknowns.

63. The Diophantine equation 2x4 y4= z2.

64. The quadratic fields K (√1), K (√ 2) and K (√3).

65. The Diophantine equation ξ3 + n3 + ζ3 = 0 and analogous equations.

66. Diophantine equations of the third degree with an infinityof solutions.

67. The Diophantine equation x7 + y7 + z7 = 0.

68, Fermat's last theorem,

69. Rational points on plane algebraic curves. Mordell's theorem.

70. Lattice points on plane algebraic curves. Theorems of Thue and Siegel.

Exercises

CHAPTER VIII: THE PRIME NUMBER THEOREM

71. Lemmata on the order of magnitude of some finite sums.

72. Lemmata on the Mohius function and some related functions.

73. Further lemmata. Proof of Selherg's formula.

74. An elementary proof of the prime number theorem.

Exercises

Table

The fundamental solutions of the equations x2 D y2 = ± 1 for D ⫹ 103.

NAME INDEX

SUBJECT INDEX

Back Cover

This is a very readable introduction to number theory, with particular emphasis on diophantine equations, and requires only a school knowledge of mathematics. The exposition is admirably clear. More advanced or recent work is cited as background, where relevant ... [T]here are welcome novelties: Gauss's own evaluation of Gauss's sums, which is still perhaps the most elegant, is reproduced apparently for the first time. There are 180 examples, many of considerable interest, some of these being little known.
Mathematical Reviews