Author(s): Trygve Nagell
Edition: 2
Publisher: Chelsea Publishing Company
Year: 1964
Language: English
City: New York
Title
Preface
Contents
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 phi-function
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 pi
Exercises
II. On the distribution of primes
14. Some lemmata
15. General remarks. The sieve of Eratosthenes
16. The function pi(x)
17. Some elementary results on the distribution of primes
18. Other problems and results concerning primes
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 prime-power modulus
28. Numerical examples of solution of algebraic congruences
29. Divisibility of integral polynomials with regard to a prime modulus
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
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 x^2 = -+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
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 - 1
51. Some trigonometrical products
52. A polynomial identity of Gauss
53. The Gaussian sums
Exercises
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 x^2 - D y^2 = 1
57. The Diophantine equation x^2 - D y^2 = -1
58. The Diophantine equation u^2 - D v^2 = C
59. Lattice points on conics
60. Rational points in the plane and on conics
61. The Diophantine equation a x^2 + b y^2 + c z^2 = 0
VII. Diophantine equations of higher degree
62. Some Diophantine equations of the fourth degree with three unknowns
63. The Diophantine equation 2 x^4 - y^4 = z^2
64. The quadratic fields K(sqrt(-1)), K(sqrt(-2)) and K(sqrt(-3))
65. The Diophantine equation xi^3 + eta^3 + zeta^3 = 0 and analogous equations
66. Diophantine equations of the third degree with an infinity of solutions
67. The Diophantine equation x^7 + y^7 + z^7 = 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
VIII. The prime number theorem
71. Lemmata on the order of magnitude of some finite sums
72. Lemmata on the Möbius function and some related functions
73. Further lemmata. Proof of Selberg’s formula
74. An elementary proof of the prime number theorem
Exercises
Tables
Name index
Subject index
