206 IV Continued Fractions and Their Uses
Thus properly equivalent forms have the same discriminant. As the name implies,
proper equivalence is indeed an equivalence relation. Moreover, any form properly
equivalent to an indefinite form is again indefinite, and any form properly equivalent
to a positive definite form is again positive definite.
We will now show that any binary quadratic form is properly equivalent to one
which is in some sense canonical. The indefinite and positive definite cases will be
treated separately.
Suppose first thatfis positive definite, so thatD<0,a>0andc>0. With the
quadratic formfwe associate a pointτ(f)of the upper half-planeH, namely
τ(f)=(−b+i√
−D)/ 2 a.Thusτ(f)is the root with positive imaginary part of the polynomialat^2 +bt+c. Con-
versely, for any givenD<0andτ∈H, there is a unique positive definite quadratic
formfwith discriminantDsuch thatτ(f)=τ.Infact,ifτ=ξ+iη,whereξ,η∈R
andη>0, we must take
a=√
(−D)/ 2 η, b=− 2 aξ, c=(b^2 −D)/ 4 a.Letf′, as above, be a form properly equivalent tof.Ift=(αt′+β)/(γt′+δ),thenat^2 +bt+c=(a′t′^2 +b′t′+c′)/(γt′+δ)^2.It follows that ifτ=τ(f)andτ′=τ(f′),thenτ=(ατ′+β)/(γτ′+δ). Thusτ′is
properly equivalent toτ, in the terminology introduced in Section 1.
By Proposition 13 we may choose the change of variables so thatτ′∈F ̄,i.e.
− 1 / 2 ≤Rτ′≤ 1 / 2 , |τ′|≥ 1.It is easily verified that this is the case if and only if forf′we have
|b′|≤a′, 0 <a′≤c′.Such a quadratic formf′is said to bereduced. Thus every positive definite binary
quadratic form is properly equivalent to a reduced form. (It is possible to ensure
that every positive definite binary quadratic form is properly equivalent to a unique
reduced form by slightly restricting the definition of ‘reduced’, but we will have no
need of this.)
If the coefficients offare integers, then so also are the coefficients off′andτ,τ′
are complex quadratic irrationals. There are only finitely many reduced formsfwith
integer coefficients and with a given discriminantD<0. For, iffis reduced, then
4 b^2 ≤ 4 a^2 ≤ 4 ac=b^2 −Dand henceb^2 ≤−D/3. Since 4ac=b^2 −D, for each of the finitely many possible
values ofbthere are only finitely many possible values foraandc.
A quadratic formf=ax^2 +bxy+cy^2 is said to beprimitiveif the coefficients
a,b,care integers with greatest common divisor 1. For any integerD<0, leth†(D)