Title | Lectures on Number Theory [electronic resource] / edited by Nikolaos Kritikos |
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Imprint | New York, NY : Springer New York, 1986 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4888-0 |
Descript | XIV, 273 p. online resource |
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. Eulerโs 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,โฆ,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 ยง32 -- 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 ยง32 -- 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 ยง46 -- 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