On the equivalence of a ternary quadratic form

Abstract

Introduction. In the classical theory of integral ternary quadratic forms a determinant a t s d= t b r $0 s r c is associated with the form f = ax2 + by2 + cz2 + 2ryz + 2sxz + 2txy. The g.c.d. of the cofactors of the elements of d is designated by Q. Then A is defined by d=Q2A. The concepts of primitive, properly primitive and reciprocal forms as well as equivalence, properties and theorems relating to such concepts are given in [1]. LEMMA 1. If g and its reciprocal G are primitive, positive or indefinite forms, then g is equivalent to a primitive form (1) f = ax2 + by2 + cz2 + 2ryz + 2sxz + 2txy in which (2) (a, t) = 1 and s # 0. PROOF. Case (i). Let g and G be properly primitive. By [2, Corollary, p. 16] g is equivalent to a form (1) whose first coefficient a and the third coefficient C of whose reciprocal form are positive integers which are relatively prime to each other and to 2Q2. Let p be an odd prime divisor of a. Then (3) (a, C) = (aC, 20A) = 1 and QC = ab t2 = p Jt. Thus (2), holds. If s=0, then replacing x by x+hz, hO, inf gives a form in which the coefficient of 2xz50 and such that a, t and C are unchanged. Case (ii). Let g be improperly primitive and G be properly primitive. By [2, Theorem 20] we may assume that a/2 and C are odd and relatively prime to each other and to OA. Thus by (3)3 QC = (a/2)(2b) t2 =* as in Case (i) p t t.