Lectures on Number Theory / edited by Nikolaos Kritikos
(Universitext)
| データ種別 | 電子ブック |
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| 出版者 | New York, NY : Springer New York |
| 出版年 | 1986 |
| 本文言語 | 英語 |
| 大きさ | XIV, 273 p : online resource |
書誌詳細を非表示
| 別書名 | 異なりアクセスタイトル:Translated from the German, with some additional material, by Schulz, William C. |
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| 内容注記 | 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 §32 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 §32 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 §46 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 |
| 一般注記 | During the academic year 1916-1917 I had the good fortune to be a student of the great mathematician and distinguished teacher Adolf Hurwitz, and to attend his lectures on the Theory of Functions at the Polytechnic Institute of Zurich. After his death in 1919 there fell into my hands a set of notes on the Theory of numbers, which he had delivered at the Polytechnic Institute. This set of notes I revised and gave to Mrs. Ferentinou-Nicolacopoulou with a request that she read it and make relevant observations. This she did willingly and effectively. I now take advantage of these few lines to express to her my warmest thanks. Athens, November 1984 N. Kritikos About the Authors ADOLF HURWITZ was born in 1859 at Hildesheim, Germany, where he attended the Gymnasium. He studied Mathematics at the Munich Technical University and at the University of Berlin, where he took courses from Kummer, Weierstrass and Kronecker. Taking his Ph. D. under Felix Klein in Leipzig in 1880 with a thes i s on modul ar funct ions, he became Pri vatdozent at Gcitt i ngen in 1882 and became an extraordinary Professor at the University of Konigsberg, where he became acquainted with D. Hilbert and H. Minkowski, who remained lifelong friends. He was at Konigsberg until 1892 when he accepted Frobenius' chair at the Polytechnic Institute in Z~rich (E. T. H. ) where he remained the rest of his 1 i fe |
| 著者標目 | Kritikos, Nikolaos editor SpringerLink (Online service) |
| 件 名 | LCSH:Mathematics LCSH:Number theory FREE:Mathematics FREE:Number Theory |
| 分 類 | DC23:512.7 |
| 巻冊次 | ISBN:9781461248880 
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| ISBN | 9781461248880 |
| URL | http://dx.doi.org/10.1007/978-1-4612-4888-0 |
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