1
1.
Preliminary comments: A quadratic form
Q(x,y)
=
ax
2
+
bxy
+
cy
2
is called
primitive
if the greatest common divisor of the coeFcients
a,b,c
is 1. If
Q
is not
primitive, it can obviously be written as
dQ
′
where
Q
′
is primitive and
d
is the
greatest common divisor of the coeFcients of
Q
. The discriminant of
Q
is
d
2
times
the discriminant of
Q
′
in this case.
(a) Show that if all the values
Q(x,y)
of a quadratic form
Q
are multiples of some
number
d >
1 then
Q
=
dQ
′
for some quadratic form
Q
′
, hence
Q
is not primitive.
Solution
:
If the values of the form
Q(x,y)
=
ax
2
+
bxy
+
cy
2
are all divisible by
d
then in particular the values
Q(
1
,
0
)
=
a
and
Q(
0
,
1
)
=
c
are divisible by
d
. Also
the value
Q(
1
,
1
)
=
a
+
b
+
c
is divisible by
d
, but since
a
and
c
are divisible by
d
,
this implies that
b
is divisible by
d
. Thus we have
Q(x,y)
=
d(
a
d
x
2
+
b
d
xy
+
c
d
y
2
)
=
dQ
′
(x,y)
.
(b) A discriminant is called
fundamental
if every quadratic form of that discriminant
is primitive. Show that a discriminant is fundamental if and only if it is not equal to
a square times the discriminant of some other form.
Solution
:
This statement is logically equivalent to the statement that a discriminant
is
not
fundamental if and only if it
is
equal to a square times the discriminant of some
other form.
In one direction: If a discriminant
Δ
is not fundamental then there exists a nonprimi-
tive form of discriminant
Δ
. This form
Q
can be written as
dQ
′
for some other form
Q
′
(where
d >
1), hence
Δ
=
d
2
Δ
′
where
Δ
′
is the discriminant of
Q
′
. Thus
Δ
is a
square times the discriminant of some other form.
In the other direction: If
Δ
is a square
d
2
times the discriminant
Δ
′
of some form
Q
′
, then the form
dQ
′
is a nonprimitive form of discriminant
Δ
, hence
Δ
is not a
fundamental discriminant.
(c) Make a list of all the discriminants between
−
50 and
+
50 that are fundamental
and another list for those that are not fundamental.
Solution
:
±irst, here are all the discriminants between
−
50 and 50:
−
48
,
−
47
,
−
44,
−
43
,
−
40
,
−
39
,
−
36
,
−
35
,
−
32
,
−
31
,
−
28
,
−
27
,
−
24
,
−
23
,
−
20
,
−
19
,
−
16
,
−
15,
−
12
,
−
11
,
−
8
,
−
7
,
−
4
,
−
3
,
0
,
1
,
4
,
5
,
8
,
9
,
12
,
13
,
16
,
17
,
20
,
21
,
24
,
25
,
28
,
29,
32
,
33
,
36
,
37
,
40
,
41
,
44
,
45
,
48
,
49. The ones that are a square times an-