CONTENTS
CHAPTER 1. BASIC CONCEPTS AND PROPOSITIONS
1. The Principle of Descent = 1
2. Divisibility and the Division A1gorithm = 3
3. Prime Numbers = 6
4. Analysis of a Composite Number into a Product of Primes = 8
5. Divisors of a Natural Number n, Perfect Numbers = 12
6. Common Divisors and Common Multiples of two or more Natural Numbers = 15
7. An Alternate Foundation of the Theory of The Greatest Common Divisor = 18
8. Euclidean Algorithm for the G.C.D. of two Natural Numbers = 21
9. Relatively Prime Natural Numbers = 23
10. Applications of the Preceding Theorems = 26
11. The Function ø(n) of Euler = 32
12. Distribution of the Prime Numbers in the Sequence of Natural Numbers = 37
Problems for Chapter 1 = 45
CHAPTER 2 CONGRUENCES
13. The Concept of Congruence and Basic Properties = 51
14. Criteria of Divisibility = 53
15. Further Theorems on Congruences = 56
16. Residue Classes mod m = 58
17. The Theorem of Fermat = 60
18. Generalized Theorem of Fermat = 61
19. Euler's Proof of the Generalized Theorem of Fermat = 62
Problems for Chapter 2 = 66
CHAPTER 3. LINEAR CONGRUENCES
20. The Linear Congruence and its Solution = 68
21. Systems of Linear Congruences = 71
22. The Case when the Moduli m1 , m2 , mk of the System of Congruences are pairwise Relatively Prime = 74
23. Decomposition of a Fraction into a Sum of An Integer and Partial Fractions = 76
24. Solution of Linear Congruences with the aid of Continued Fractions = 79
Problems for Chapter 3 = 84
CHAPTER 4. CONGRUENCES OF HIGHER DEGREE
25. Generalities for Congruence of Degree k > 1 and Study of the Case of a Prime Modulus = 89
26. Theorem of Wilson = 93
27. The System {r, r2 ,} of Incongruent Powers Modulo a prime p = 94
28. Indices = 96
29. Binomial Congruences = 99
30. Residues of Powers Mod p = 101
31. Periodic Decadic Expansions = 103
Problems for Chapter 4 = 106
CHAPTER 5. QUADRATIC RESIDUES
32. Quadratic Residues Modulo m = 109
33. Criterion of Euler and the Legendre Symbol = 109
34. Study of the Congruence x2 ≡ a (mod pr = 112
35. Study of the Congruence x2 ≡ a (mod 2k = 116
36. Study of the Congruence x2 ≡ a (mod m) with (a,m) ＝ l = 120
37. Generalization of the Theorem of Wilson = 123
38. Treatment of the Second Problem of §32 = 127
39. Study of (-1／p) and Applications = 128
40. The Lemma of Gauss = 129
41. Study of (2／p) and an application = 133
42. The Law of Quadratic Reciprocity = 135
43. Determination of the 0dd Primes p for which (q／p) ＝ 1 with given q = l38
44. Generalization of the Symbol (a／p) of Legendre by Jacobi = 139
45. Completion of the Solution of the Second Problem of §32 = 146
Problems for Chapter 5 = 151
CHAPTER 6. BINARY QUADRATIC FORMS
46. Basic Notions = 157
47. Auxiliary Algebraic Forms = 160
48. Linear Transformation of the Quadratic Form a x2 + 2bxy + c y2 = 161
49. Substitutions and Computation with them = 162
50. Unimodular Transformations (or Unimodular Substitutions) = 168
51. Equivalence of Quadratic Forms = 170
52. Substitutions Parallel to ( 0 -1 1 0 ) = 172
53. Reductions of the First Basic Problem of §46 = 174
54. Reduced Quadratic Forms with Discriminant △ < 0 = 178
55. The Number of Classes of Equivalent Forms with Discriminant △ < 0 = 184
56. The Roots of a Quadratic Form = 187
57. The Equation of Fermat (and of Pell and Lagrange) = 192
58. The Divisors of a Quadratic Form1 = 98
59. Equivalence of a form with itself and solution of the Equation of Fermat for Forms with Megative Discriminant △ = 201
60. The Primitive Representations of an odd Integer by x2 + y2 = 203
61. The Representation of an Integer m by a Complete System of Forms with given Discriminant △ < 0 = 205
62. Regular Continued Fractions = 213
63. Equivalence of Real Irrational Numbers = 219
64. Reduced Quadratic Forms with Discriminant △ > 0 = 226
65. The Period of a Reduced Quadratic Form With △ > 0 = 232
66. Development of Δ in a Continued Fraction = 241
67. Equivalence of a form with itself and solution of the equation of Fermat for forms with Positive Discriminant △ = 243
Problems for Chapter 6 = 252
BIBLIOGRAPHY = 265
INDEX = 272
