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Fraction Factorization **Previous:** Theorems of Special Interest

**Lemma 2** : Let be a square-free
integer, be the greatest integer in , as defined in (4) and
. Then if and only if

**Proof:** For a proof, see [??].

**Theorem 3** (B = 2C) :
Let be a
square-free integer such that the continued fraction
representation of has a center b, be the greatest integer in
, , and be defined as in (4). Then if and only
if .

**Proof:** Suppose that . Then so by Lemma 2,

which leads to

Carrying out this fractional linear transformation yields

Substituting in for and making the
substitution stated in (8) gives
the equation

But by the construction of , det =. Thus making the
substitution into the above equation,
we get the quadratic equation (in B)

Solving for yields

But by the choice of and the construction of and , we have that and thus the second possibility must be
ruled out because it would yield a negative value for . Hence as
required. Conversly, assume that . Then it is true that

But once again, so if we make the
substitution , it is true that

Then making the substitution and
the one stated in (8) yields (after
rearranging terms)

It follows that

and thus by Lemma 2, and .

For the remainder of the report we consider those
continued fractions such that . Then we can make a change
of variables in the matrix in (4) and
let

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**Up:** Continued
Fraction Factorization **Previous:** Theorems of Special Interest