Lectures on Number Theory

1. Basic Concepts and Propositions.- 1. The Principle of Descent.- 2. Divisibility and the Division Algorithm.- 3. Prime Numbers.- 4. Analysis of a Composite Number into a Product of Primes.- 5. Divisors of a Natural Number n, Perfect Numbers.- 6. Common Divisors and Common Multiples of two or more Natural Number.- 7. An Alternate Foundation of the Theory of The Greatest Common Divisor.- 8. Euclidean Algorithm for the G.C.D. of two Natural Numbers.- 9. Relatively Prime Natural Numbers.- 10. Applications of the Preceding Theorems.- 11. The Function ?(n)of Euler.- 12. Distribution of the Prime Numbers in the Sequence of Natural Numbers.- Problems for Chapter 1.- 2. Congruences.- 13. The Concept of Congruence and Basic Properties.- 14. Criteria of Divisibility.- 15. Further Theorems on Congruences.- 16. Residue Classes mod m.- 17. The Theorem of Fermat.- 18. Generalized Theorem of Fermat.- 19. Euler's Proof of the Generalized Theorem of Fermat.- Problems for Chapter 2.- 3. Linear Congruences.- 20. The Linear Congruence and its Solution.- 21. Systems of Linear Congruence.- 22. The Case when the Moduli $${m_1},{m_2}, ldots ,{m_k}$$ of the System of Congruences are pairwise Relatively Prime.- 23. Decomposition of a Fraction into a Sum of An Integer and Partial Fractions.- 24. Solution of Linear Congruences with the aid of Continued Fractions.- Problems for Chapter 3.- 4. Congruences of Higher Degree.- 25. Generalities for Congruence of Degree k >1 and Study of the Case of a Prime Modulus.- 26. Theorem of Wilson.- 27. The System {r,r2,...,r?} of Incongruent Powers Modulo a prime p.- 28. Indices.- 29. Binomial Congruences.- 30. Residues of Powers Mod p.- 31. Periodic Decadic Expansions.- Problems for Chapter 4.- 5. Quadratic Residues.- 32. Quadratic Residues Modulo m.- 33. Criterion of Euler and the Legendre Symbol.- 34. Study of the Congruence X2 ? a (mod pr).- 35. Study of the Congruence X2 ? a (mod 2k).- 36. Study of the Congruence X2 ? a (mod m) with (a,m)=1.- 37. Generalization of the Theorem of Wilson.- 38. Treatment of the Second Problem of x32.- 39. Study of $$left( {frac{{ - 1}}{p}} right)$$ and Applications.- 40. The Lemma of Gauss.- 41. Study of $$left( {frac{2}{p}} right)$$ and an application.- 42. The Law of Quadratic Reciprocity.- 43. Determination of the Odd Primes p for which $$left( {frac{q}{p}} right) = 1$$ with given q.- 44. Generalization of the Symbol $$left( {frac{a}{p}} right)$$ of Legendre by Jacobi.- 45. Completion of the Solution of the Second Problem of x32.- Problems for Chapter 5.- 6. Binary Quadratic Forms.- 46. Basic Notions.- 47. Auxiliary Algebraic Forms.- 48. Linear Transformation of the Quadratic Form ax2 + 2bxy + cy2.- 49. Substitutions and Computation with them.- 50. Unimodular Transformations (or Unimodular Substitutions).- 51. Equivalence of Quadratic Forms.- 52. Substitutions Parallel to $$left( {begin{array}{*{20}{c}} 0&{ - 1} 1&0 end{array}} right)$$.- 53. Reductions of the First Basic Problem of x46.- 54. Reduced Quadratic Forms with Discriminant ? < 0.- 55. The Number of Classes of Equivalent Forms with Discriminant ? < 0.- 56. The Roots of a Quadratic Form.- 57. The Equation of Fermat (and of Pell and Lagrange).- 58. The Divisors of a Quadratic Form.- 59. Equivalence of a form with itself and solution of the Equation of Fermat for Forms with Negative Discriminant ?.- 60. The Primitive Representations of an odd Integer by x2+y2.- 61. The Representation of an Integer m by a Complete System of Forms with given Discriminant ? < 0.- 62. Regular Continued Fractions.- 63. Equivalence of Real Irrational Number.- 64. Reduced Quadratic Forms with Discriminant ? < 0.- 65. The Period of a Reduced Quadratic Form With ? < 0.- 66. Development of $$sqrt Delta $$ in a Continued Fraction.- 67. Equivalence of a form with itself and solution of the equation of Fermat for forms with Positive Discriminant ?.- Problems for Chapter 6.