| This dissertation discusses the regular continued fraction expansion of DX where D(X) is a quadratic polynomial whose coefficients satisfy a certain divisibility condition.;There are two main results in this dissertation. We show that certain families of non-square D may be represented by some quadratic D(X) satisfying the divisibility condition and give a surprising period length property of the product of some members of a family. The second result generalizes the work of van der Poorten and Williams [206] concerning the continued fraction expansion of DX for sufficiently large X, where D( X) obeys the divisibility condition. Also, we establish an upper bound for the period length of the continued fraction expansion of using the Lucas-Lehmer theory, and construct the fundamental unit of the real quadratic order [1, DX ]. |
