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Fraction Factorization **Previous:** Proof of Main Theorem

We are interested in the factorization

for and . Once this matrix factorization is accomplished, we see that the required continued fraction factorization of into is simply .

Let us look again at the examples from section 2:

**Ex.1:**

**Ex.2:**

**Ex.3:**

The first two matrix factorizations agree with
the factorizations given in section 2. The third example does not seem
to agree. However, calculating such things as [5/8] and [5/16],
we see that there is still a correspondence between the matrix
factorization and the continued fraction factorization. In fact,
if we were to carry out the matrix factorization in the following
form

we see that we can factor the matrix for as

in which case the matrix factorization does agree with the factorization given in section 2. This leads to the numerous cases that must be considered when considering continued fraction factorizations. Other cases occur when or when in (1), both of which require new techniques when considering factorization.

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Fraction Factorization **Previous:** Proof of Main Theorem