Prime Numbers

240 Chapter 5 EXPONENTIAL FACTORING ALGORITHMS
Allowing both +1 and −1 for the determinant does not give much more
leeway than restricting to just +1. (For example, one can go from (a, b, c)
to (a, −b, c) and to (c, b, a) via changes of variables with determinants −1,
but these are easily recognized, and may be tacked on to a more complicated
change of variables with determinant +1, so there is little loss of generality
in just considering +1.) We shall say that two quadratic forms are equivalent
if there is a change of variables as in (5.1) with determinant +1. Such a
change of variables is called unimodular, and so two quadratic forms are called
equivalent if you can go from one to the other by a unimodular change of
variables.
Equivalence of quadratic forms is an “equivalence relation.” That is, each
form (a, b, c) is equivalent to itself; if (a, b, c) is equivalent to (a ′ ,b ′ ,c ′ ), then
the reverse is true, and two forms equivalent to the same form are equivalent
to each other. We leave the proofs of these simple facts as Exercise 5.10.
There remains the computational problem of deciding whether two given
quadratic forms are equivalent. The discriminant of a form (a, b, c) isthe
integer b 2 − 4ac. Equivalent forms have the same discriminant (see Exercise
5.12), so it is sometimes easy to see when two quadratic forms are not
equivalent, namely this is so when their discriminants are unequal. However,
the converse is not true. Witness the two forms x 2 +xy+4y 2 and 2x 2 +xy+2y 2 .
They both have discriminant −15, but the first can have the value 1 (when
x = 1 and y = 0), while the second cannot. So the two forms are not
equivalent.
If it is the case that in each equivalence class of binary quadratic forms
there is one distinguished form, and if it is the case that it is easy to find
this distinguished form, then it will be easy to tell whether two given forms
are equivalent. Namely, find the distinguished forms equivalent to each, and
if these distinguished forms are the same form, then the two given forms are
equivalent, and conversely.
This is particularly easy to do in the case of binary quadratic forms of
negative discriminant. In fact, the whole theory of binary quadratic forms
bifurcates on the issue of the sign of the discriminant. Forms of positive
discriminant can represent both positive and negative values, but this is not
the case for forms of negative discriminant. (Forms with discriminant zero are
trivial objects—studying them is essentially studying the sequence of squares.)
The theory of binary quadratic forms of positive discriminant is somewhat
more difficult than the corresponding theory of negative-discriminant forms.
There are interesting factorization algorithms connected with the positivediscriminant
case, and also with the negative-discriminant case. In the
interests of brevity, we shall mainly consider the easier case of negative
discriminants, and refer the reader to [Cohen 2000] for a description of
algorithms involving quadratic forms of positive discriminant.
We make a further restriction. Since a binary quadratic form of negative
discriminant does not represent both positive and negative numbers, we shall
restrict attention to those forms that never represent negative numbers. If
(a, b, c) is such a form, then (−a, −b, −c) never represents positive numbers,