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By Ionin Y. J., Shrikhande M. S.

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In particular, there is an especially nice notion of reduced form. 7) \b\ < a < c, and b > 0 if either \b\ = a or a - c. 8. Every primitive positive definite form is properly equivalent to a unique reduced form. Proof. The first step is to show that a given form is properly equivalent to one sat­ isfying \b\ < a < c. Among all forms properly equivalent to the given one, pick f(x,y) = ax2 + bxy + cy2 so that \b\ is as small as possible. If a < \b\, then g(x,y) = f(x + my,y) = ax2 + (2am + b)xy + c'y2 is properly equivalent to f(x,y) for any integer m.

But the uniqueness problem is much more complicated. As Gauss notes, "it can happen that many reduced forms are properly equivalent among themselves" [41, §184]. Determining exactly which reduced forms are prop­ erly equivalent is not easy (see Lagrange [69, pp. 728-740] and Gauss [41, §§183— 193]). There are also connections with continued fractions and Pell's equation (see [41, §§183-205]), so that the indefinite case has a very different flavor. Two modern references are Flath [36, Chapter IV] and Zagier [111, §§8, 13 and 14].

307]: Lagrange is thefirstwho opened the way for the study of these sorts of theo­ rems. But the methods which served the great geometer are not applicable ... D. LAGRANGE AND LEGENDRE 37 except in very few cases; and the difficulty in this regard could not be completely resolved without the aid of the law of reciprocity. Besides completing Lagrange's program, Legendre also tried to understand some of the other ideas implicit in Lagrange's memoir. We will discuss one of Legendre's attempts that is particularly relevant to our purposes: his theory of composition.

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