Motivation and Examples

Here is a Java applet I wrote to compute the continued fraction expansion for the square roots of positive integers: Continued Fraction Applet.

Consider the following examples:

Ex.1: $d=118$, $r=\frac{7}{6}$

$$\sqrt{118}=[10;\overline{1,6,3,2,10,2,3,6,1,20}]$$

$$[1,6,3,2]=[1,6][3,2]=\left[\frac{7}{6}\right]\left[\overline{\frac{7}{3}}\right]$$

Ex.2: $d=162$, $r=\frac{3}{2}$

$$\sqrt{162}=[12;\overline{1,2,1,2,12,2,1,2,1,24}]$$

$$[1,2,1,2]=[1,2][1,2]=\left[\frac{3}{2}\right][\overline{3}]$$

Ex.3: $d=179$, $r=\frac{8}{3}$

$$\sqrt{179}=[13;\overline{2,1,1,1,3,5,13,5,3,1,1,1,2,26}]$$

$$[2,1,1,1,3,5]=[2,1,1,1][3,5]=\left[\frac{8}{3}\right]\left[\overline{\frac{16}{3}}\right]$$

We see in these examples that the factorization is indeed possible. These factorizations can be accomplished by trial and error but methods which are discussed in these report make the factorization easier to accomplish. These methods involve the correspondence between continued fractions and matrices.