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.

Table of Contents:
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 ContinuedFractions.- 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.