there is a unique reduced form of discriminant -4⁢n only for n=1,2,3,4,7

there is a unique reduced form of discriminant

-

4

⁢

n

only for

n

=

1

,

2

,

3

,

4

,

7

The number of reduced

http://planetmath.org/node/IntegralBinaryQuadraticForms

integral binary quadratic forms

of a given

http://planetmath.org/node/IntegralBinaryQuadraticForms

discriminant

Δ

<

0

is finite; the number of such forms is

h

Δ

.

Theorem 1

.

Let

n

be a

positive integer

. Then

h

-

4

⁢

n

=

1

if and only if

n

=

1

,

2

,

3

,

4

,

or

⁢

7

.

Proof.

(This proof is taken from

[

1

]

, which is itself taken from an earlier proof).

By computing all

reduced forms

of discriminants

-

4

,

-

8

,

-

12

,

-

16

,

-

20

one can see that

h

-

4

⁢

n

=

1

in the five cases given in the statement of the theorem. We show there are no others.

Clearly

x

2

+

n

⁢

y

2

is a reduced form of discriminant

-

4

⁢

n

. For

n

∉

{

1

,

2

,

3

,

4

,

7

}

, we will produce a second reduced form of the same discriminant, showing that

h

-

4

⁢

n

>

1

. We may assume

n

>

1

, since we already know that

h

-

4

=

1

.

Suppose first that

n

has at least two

distinct prime factors

. Then we can write

n

=

a

⁢

c

where

1

<

a

<

c

,

gcd

⁡

(

a

,

c

)

=

1

. Then

a

⁢

x

2

+

c

⁢

y

2

is reduced, and its discriminant is

-

4

⁢

a

⁢

c

=

-

4

⁢

n

. So if

n

has two distinct prime factors,

h

-

4

⁢

n

>

1

.

We now consider the prime power case, taking

2

r

and

p

r

,

p

an

odd prime

, separately.

If

n

=

2

r

, then we already know that for

r

=

1

,

2

,

h

-

4

⁢

n

=

1

. For

r

=

3

, one can compute the classes of discriminant

-

32

and see that

h

-

4

⁢

n

=

2

. For

r

≥

4

, then

4

⁢

x

2

+

4

⁢

x

⁢

y

+

(

2

r

-

2

+

1

)

⁢

y

2

is clearly

primitive

, and is also reduced since

4

≤

2

r

-

2

+

1

. Further, its discriminant is

4

2

-

4

⋅

4

⋅

(

2

r

-

2

+

1

)

=

-

16

⋅

2

r

-

2

=

-

4

⁢

n

. Thus in this case as well,

h

-

4

⁢

n

>

1

.

Finally, suppose

n

=

p

r

,

p

an odd prime. Suppose we can write

n

+

1

=

a

⁢

c

,

2

≤

a

<

c

,

gcd

⁡

(

a

,

c

)

=

1

. Then

a

⁢

x

2

+

2

⁢

x

⁢

y

+

c

⁢

y

2

is reduced and has discriminant

-

4

⁢

n

, so

h

-

4

⁢

n

>

1

. So we are left with the case where

n

+

1

is a prime power which, since it is even, must be

2

s

.

s

=

1

,

2

,

3

correspond to

n

=

1

,

3

,

7

;

s

=

4

corresponds to

n

=

15

, which is not a prime power; and for

s

=

5

, one can simply compute the forms of discriminant

-

4

⋅

31

to see that

h

-

4

⋅

31

=

3

. So the only possibility remaining is that

s

≥

6

. In this case, though,

8

⁢

x

2

+

6

⁢

x

⁢

y

+

(

2

s

-

3

+

1

)

⁢

y

2

has relatively prime

coefficients

, and is reduced since

s

≥

6

⇒

8

≤

2

s

-

3

+

1

. Also, its discriminant is

6

2

-

4

⋅

8

⋅

(

2

s

-

3

+

1

)

=

4

-

4

⋅

2

s

=

4

-

4

⁢

(

n

+

1

)

=

-

4

⁢

n

, and thus

h

-

4

⁢

n

>

1

in this case as well.
∎

References

1

Cox, D.A.

Primes of the Form

x

2

+

n

⁢

y

2

: Fermat,

Class Field Theory

, and

Complex Multiplication

, Wiley 1997.

Title

there is a unique reduced form of discriminant

-

4

⁢

n

only for

n

=

1

,

2

,

3

,

4

,

7

Canonical name

ThereIsAUniqueReducedFormOfDiscriminant4nOnlyForN12347

Date of creation

2013-03-22 16:56:46

Last modified on

2013-03-22 16:56:46

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 11E12

Classification

msc 11R29

Classification

msc 11E16

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