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Consider the field
Looking at the determinant of
we see that and thus . Let . Then . It is clear that there seems to be a correspondence between and . Thus one could consider the factorization of by considering the factorizatino of in .
A unit in is a number of the form . A
number in is said to be prime if implies that or is a unit. Let be such numbers. Then we can look at the
This correspondence between prime factorization in and the factorization of continued
fractions could be an area for further research.