On quasi-reduced quadratic forms

Abstract

With the help of continued fractions, we plan to list all the elements of the set $$ \mathcal{Q}_\Delta = \{ aX^2 + bXY + cY^2 :a,b,c \in \mathbb{Z},b^2 - 4ac = \Delta with0 \leqslant b < \sqrt \Delta \} $$ of quasi-reduced quadratic forms of fundamental discriminant Δ. As a matter of fact, we show that for each reduced quadratic form f = aX 2 + bXY + cY 2 = 〈a, b, c〉 of discriminant Δ > 0 (and of sign σ(f) equal to the sign of a), the quadratic forms associated with f and defined by $$ \left\{ \begin{gathered} \left\langle {a + bu + cu^2 ,b + 2cu,c} \right\rangle ,with1 \leqslant \sigma (f)u \leqslant \frac{b} {{2|c|}}(whenevertheyexist), \hfill \\ \left\langle {c, - b - 2cu,a + bu + cu^2 } \right\rangle ,with\frac{b} {{2|c|}} \leqslant \sigma (f)u \leqslant [\omega (f)] = \left[ {\frac{{b + \sqrt \Delta }} {{2|c|}}} \right], \hfill \\ \end{gathered} \right. $$ are all different from one another and build a set I(f) whose cardinality is $$ \# I(f) = \left\{ \begin{gathered} 1 + [\omega (f)],when(2c)|b, \hfill \\ [\omega (f)],when(2c)\not |b. \hfill \\ \end{gathered} \right. $$ If f and g are two different reduced quadratic forms, we show that I(f) ∩ I(g) = $$ \not 0 $$ . Our main result is that the set Q Δ is given by the disjoint union of all I(f) with f running through the set of reduced quadratic forms of discriminant Δ > 0. This allows us to deduce a formula for #(Q Δ) involving sums of partial quotients of certain continued fractions.