A description of continued fraction expansions of quadratic surds represented by polynomials

AbstractThe principal thrust of this investigation is to provide families of quadratic polynomials {Dk(X)=fk2X2+2ekX+C}k∈N, where ek2−fk2C=n (for any given nonzero integer n) satisfying the property that for any X∈N, the period length ℓk=ℓ(Dk(X)) of the simple continued fraction expansion of Dk(X) is constant for fixed k and limk→∞ℓk=∞. This generalizes, and completes, numerous results in the literature, where the primary focus was upon |n|=1, including the work of this author, and coauthors, in Mollin (Far East J. Math. Sci. Special Vol. 1998, Part III, 257–293; Serdica Math. J. 27 (2001) 317) Mollin and Cheng (Math. Rep. Acad. Sci. Canada 24 (2002) 102; Internat Math J 2 (2002) 951) and Mollin et al. (JP J. Algebra Number Theory Appl. 2 (2002) 47)