Theory of Numbers Volume 1 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1892 Excerpt: ...are numbers representable by it; and if (a, b, c) is properly primitive, one or other of the numbers a, c must be prime to any given prime factor of D, and one or other of them must be odd. All the particular characters of a properly primitive form may therefore be determined from its extreme coefficients. For example, let the form be (6, 3, 13). Here D =-69 =-3.23a3 (mod 4), and the particular characters are (w 3), (n23), and %. From the coefficient 13, which is odd and prime to 3, we obtain x = (-- ). = +, (W3) = (133) = + l; and from the coefficient 6, we find (n23) = (6I23) = (223) (323) = (+ 1)(+1) = + 1.' Therefore the total character of the form is (-8)-+ 1, (w23) = + l, X = + l; or, as Gauss would express it, 1, 4; RS, B2S. 132. In the case of improperly primitive forms, the characters will be as in the first line of Dirichlet's table, except that n will denote an odd number the double of which is represented by the form. Here again, the total character may be assigned by inspection of the extreme coefficients of the form. Thus, if the form be (10, 5, -4), for which D = 65 = 5.13, the particular characters are (n 5) and (w 13). Putting n =--4/2 =--2, we have (n 5) = (-2I5) =-1, (w13) = (-2I13) =-1. 'The generic characters of derived forms may be at once inferred from those of the primitive forms from which they are derived. 133. The following tables are given by way of further illustration. For the negative determinants, representatives of the positive primitive classes only have been given. Each line of a table gives the total character of a genus, and representatives of all the classes belonging to that genus. Improperly primitive genera, when they exist, are separated from the rest by a horizontal line. Derived genera are omitted. D =-96. 134...