Introduction to number theory

Title......Page 1
PREFACE......Page 3
CONTENTS......Page 5
I. Divisors......Page 9
2. Remainders......Page 10
3. Primes......Page 11
4. The fundamental theorem......Page 12
5. Least common multiple and greatest common divisor......Page 14
6. Moduls, rings and fields......Page 17
7. Euclid\'s algorithm......Page 19
8. Relatively prime numbers. Euler\'s q-function......Page 21
9. Arithmetical functions......Page 24
10. Diophantine equations of the first degree......Page 27
11. Lattice points and point lattices......Page 30
12. Irrational numbers......Page 32
13. Irrationality of the numbers e and T......Page 36
Exercises......Page 38
14. Some lemmata......Page 45
15. General remarks. The sieve of Eratosthenes......Page 49
16. The function pi (x).......Page 52
17. Some elementary results on the distribution of primes......Page 55
18. Other problems and results concerning primes......Page 62
19. Definitions and fundamental properties......Page 66
20. Residue classes and residue systems......Page 67
21. Fermat\'s theorem and its generalization by Euler......Page 69
22. Algehraic congl\"Uences and functional congruences......Page 71
23. Linear congruences......Page 74
24. Algebraic cougruences to a prime modulus......Page 77
25. Prime divisors of integral polynomials......Page 79
26. Algebraic congruences to a composite modulus.......Page 81
27. Algebraic congruences to a prime-power modulus......Page 83
28. Numerical examples of solution of algebraic congruences......Page 88
29. Divisibility of integral polynomials with regard to a primemodulus......Page 91
30. Wilson\'s theorem and its generalization......Page 97
31. Exponent of an integer modulo n......Page 100
32. Moduli having primitive roots......Page 105
33. The index calculus......Page 109
34. Power residues. Binomial congruences......Page 113
35. Polynomials representing integers......Page 118
36. Thue\'s remainder theorem and its generalization by Scholz......Page 120
Exercises......Page 122
37. The general quadratic congruence......Page 130
38. Euler\'s criterion and Legendre\'s symbol......Page 131
39. On the solvability of the congruences......Page 134
40. Gauss\'s lemma......Page 137
41. The quadratic reciprocity law......Page 139
42. Jacobi\'s symbol and the generalization of the reciprocity law......Page 143
43. The prime dh-isors of quadratic polynomials......Page 147
44. Primes in special arithmetical progressions......Page 151
45. The roots of unity......Page 154
46. The cyclotomic polynomial......Page 156
47. Irreducibility of the cyclotomic polynomial......Page 158
48. The prime divisors of the cyclotomic polynomial......Page 162
49. A theorem of Bauer on the prime divisors of certain polynomials......Page 166
50. On the primes of the form ny - 1......Page 168
51. Some trigonometrical products......Page 171
52. A polynomial identity of Gauss......Page 172
53. The Gaussian sums......Page 175
Exercises......Page 178
54. The representation of integers as sums of integral squares......Page 186
55. Bachet\'s theorem......Page 189
56. The Diophantine equation x^2 - Dy^2 = 1.......Page 193
57. The Diophantine equation x^2 - Dy^2 = -1.......Page 199
58. The Diophantine equation u^2 - D v^2 = C.......Page 202
59. Lattice points on conics......Page 210
60. Rational points in the plane and on conics......Page 214
61. The Diophantine equation ax^2 + by^2 + cz^2 =0.......Page 216
62. Some Diophantine equations of the fourth degree with three unknowns......Page 225
63. The Diophantine equation 2x^4 - y^4 = z^2......Page 230
64. The quadratic fields......Page 233
65. The Diophantine equation x^3 + y^3 + z^3 = 0 and analogous equations......Page 239
66. Diophantine equations of the third degree with an infinity of solutions......Page 244
67. The Diophantine equation x^7 + y^7 + z^7 = O......Page 246
68. Fermat\'s last theorem......Page 249
69. Rational points on plane algebraic curves.Mordell\'s theorem......Page 251
70. Lattice points on plane algebraic curves. Theorems of Thue and Siegel......Page 258
Exercises......Page 263
71. Lemmata on the order of magnitude of some finite sums......Page 273
72. Lemmata on the Mobius function and some related functions......Page 276
73. Further lemmata. Proof of Selberg\'s formula......Page 281
74. An elementary proof of the prime number theorem......Page 284
Exercises......Page 296
Table......Page 298
NAME INDEX......Page 300
SUBJECT INDEX......Page 302