Factorization

We are interested in the factorization

$$\left(\begin{array}{cc} A & 2B\\ B & C \end{array}... ...begin{array}{cc} s & t\\ 2u & v \end{array} \right)^T$$ (11)

for $s,t,u,v \in \mathbb{Z}$ and $AC-2B^2=1$. Once this matrix factorization is accomplished, we see that the required continued fraction factorization of $[a_1,a_2,\ldots,a_n]$ into $[r][\overline{2r}]$ is simply $\left[\frac{v}{2t}\right]\left[\overline{\frac{v}{t}}\right]$.

Let us look again at the examples from section 2:

Ex.1: $d=118$

$$\left(\begin{array}{cc} 19 & 44\\ 22 & 51 \end{arra... ...\begin{array}{cc} 1 & 3\\ 2 & 7 \end{array} \right)^T$$

Ex.2: $d=162$

$$\left(\begin{array}{cc} 3 & 8\\ 4 & 11 \end{array} ... ...\begin{array}{cc} 1 & 1\\ 2 & 3 \end{array} \right)^T$$

Ex.3: $d=179$

$$\left(\begin{array}{cc} 11 & 58\\ 29 & 153 \end{arr... ...begin{array}{cc} 3 & 1\\ 16 & 5 \end{array} \right)^T$$

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

$$\left(\begin{array}{cc} A & 2B\\ B & C \end{array}... ...egin{array}{cc} s & t\\ u & 2v \end{array} \right)^T,$$ (12)

we see that we can factor the matrix for $d=179$ as

$$\left(\begin{array}{cc} 11 & 58\\ 29 & 153 \end{arr... ...begin{array}{cc} 1 & 3\\ 5 & 16 \end{array} \right)^T$$

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 $AC-2B^2=-1$ or when $a_0 \ne b$ in (1), both of which require new techniques when considering factorization.