Binary quadratic form is one of the subjects treated in elementary number theory. Another subject treated in elementary number theory is the possibility of representing a positive integer as a sum of two squares and difference of two squares. The representations

Let

If

This group can be presented as

Now the product of two transformations is the same as the product of corresponding matrices. For the sake of simplicity, we use matrices instead of transformations.

A coset diagram is a graph consisting of vertices and edges. It depicts a permutation representation of the modular group G, the 3-cycles of

In ^{1, 2}, types of length 4, 6 satisfying exactly one of the conditions namely

In ^{3, 4} formula for total numbers of ambiguous numbers in ^{5} it is explored that if ^{6} it is describe that if

^{7} Let

^{8} If a natural number

^{9} Any two elements of the same order are conjugate in a group G.

^{6}

For

In this section we determine the elements of G and conditions on

In the following theorem, we describe the elements of G that moves real quadratic irrational numbers to their conjugates.

That is

This implies that

This can be written as

This gives

So, we have g =

Then

Since g is an element of order 2, but any two elements of same order are conjugate by lemma 2.3. So, g is of the form

But both elements can be written as

Conversely, if

^{5}, that

As

Now

^{5}, that

Also

On contrary, we suppose that

Then,

That is,

Thus

In the following theorem we determine condition on

Consider

This implies that,

That is,

Consider

That is

After simplification, we have

After rationalization, we have

This can be written as

After simplification, we have

Following corollary is an immediate consequence of the above result.

Now

Similarly, for

As

In ^{1, 2} types of lengths 4, 6 have been determined in which all the four orbits

The following corollary follows from theorem 3.2 and corollary 3.2.

Now

Converse of above result is not hold because

But the orbit does not contain these ambiguous numbers

It has been proved in ^{5}, that

That is

This implies that

This can be written as

This gives

Combining both equations, we have

After simplification, we obtain

But

By substitution, we have

This can be written as

After substituting, the value of

After simplification, we obtain

This can be written as

In this expression

By Lemma 2.2 if a natural number

Let

This shows that

This implies that

Also,

This implies that

Thus

In the orbit

Now corresponding element in matrix form is given by:

Here

Now

After substituting the values of

As required.

In the following theorem, we generalize the results of ^{5}. In particular, we describe the condition on

That is

This implies that

This can be written as

Which gives

Combining both equations, we have

After simplification, we obtain

This implies that

By substituting the value of

After substituting the value of

After some simplification, we have

This can be written as

In this expression

By Lemma 2.2 if a natural number

Let

This shows that

This implies that

Also,

This implies that

Thus

In the orbit

Now corresponding element in matrix form is given by:

Here

Now

After substituting the values of

As required.

That is

This implies that

This can be written as

This gives

Combining both equations, we have

After simplification, we obtain

But

This can be written as

After substituting, the value of

After simplification, we obtain

This can be written as

If

In the similar way, by eliminating s and u we can obtain

Now corresponding element in matrix form is given by

Here

Now

After substituting the values of

The element of G which moves

Now corresponding element in matrix form is given by

Here

Now

After substituting the values of

As required.

The idea of study the elements that moves