having the value non-residue of p. +1, if a is a residue of p, and -1, if a is a Then we have always: 57. To determine whether or not any number m is a residue of p, it is sufficient to determine whether or not -1, 2, and the odd prime factors of m are residues of p. The following results may be proved: 58. The last is an important theorem, known as Legendre's Law of Reciprocity, and may be stated as follows: If p and q are two odd primes and if at least one of them is of the form 4n +1, then q is a residue of p, if and only if p is a residue of q, while if both p and q are of the form 4n+3, then q is a residue of p when p is a non-residue of q, and vice versa. This theorem was discovered empirically by Euler (1783), announced in its general form by Legendre (1785), and partly proved by him. The first complete proof was, however, due to Gauss, who gave eight distinct proofs. Many others have been given down to the present time.* For further information and for a full presentation of some of the proofs, the reader is referred to the works mentioned in the Bibliography. * A chronological list of 49 proofs, extending from the first proof, published by Gauss in 1801, to three proofs by Lange in 1896–97, is given in Bachmann, Niedere Zahlentheorie, I, pp. 203-4. VII. BIBLIOGRAPHY 59. The classic work in our subject is the Disquisitiones Arithmetica of C. F. Gauss, published in 1801, when Gauss was only twenty-four years of age, and really completed a few years earlier. In this work Gauss gave a masterly presentation of the subject which has remained unequalled; unlike many masterpieces, it is written so clearly and simply that much of it is intelligible to the beginner. A German translation by Maser (Berlin, 1889), and a French translation by Poullet-Delisle (Paris, 1807), make the work more widely accessible. The following texts also take up the subject from the beginning, reaching varying degrees of advancement: Dirichlet-Dedekind, Zahlentheorie, Braunschweig, 4th ed., 1894. Cahen, Théorie des Nombres, Paris, 1900. Mathews, Theory of Numbers, I, Cambridge, 1892. These works contain numerous references, both to the older and the contemporary literature. An excellent sketch of the principal results and present state of our subject is given in the Encyclopädie der Mathematischen Wissenschaften, Band I, 2ter Teil, appearing with additions in the French translation, Encyclopédie des Sciences Mathématiques, Tome I, Vol. III. The theory of numbers figures largely in the field of "Mathematical recreations." An introduction to this field may be obtained through some or all of the following: Ball, Mathematical Recreations and Problems, 3d ed., London, 1890. Bachet de Méziriac, Problèmes plaisants et délectables qui se font par les nombres. First published in 1612, and reprinted at Paris in 1884. Lucas, Récréations Mathématiques, 4 vols., Paris, 1891–96. Ahrens, Mathematische Unterhaltungen und Spiele, Leipzig, 1900. In this connection mention may also be made of a paper by Bouton on "Nim, A Game with a Complete Mathematical Theory" (Annals of Math., ser. 2, Vol. III, pp. 35-39, 1901), recently generalized by Moore (ibid., Vol. XI, pp. 90-94, 1910). CONTENTS 1. Introduction. 2. Analytic criterion for constructibility. 3. Graphical solution of a quadratic equation. 4. Domain of rationality. 5. Functions involving no irrationalities other than square root. 9. Reducible and irreducible functions. 11. Fundamental theorem; Duplication of the cube; Trisection of an angle; Quadrature of the circle. 13. Connection between regular polygons and roots of unity. 14. De Moivre's theorem. 17. Regular pentagon and decagon. 19. Regular polygon of 17 sides. 20. Construction of the regular polygon of 17 sides. 21. Gauss's theory of regular polygons. 28. Primitive roots of unity. 30. Gauss's lemma. 31. Irreducibility of the cyclotomic equation. 32. Proofs of theorems cited earlier. 39. References. 352 |