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

Abstract

The principal thrust of this investigation is to provide families of quadratic polynomials {D k(X)=f k 2X 2+2e kX+C} k∈ N , where e k 2− f k 2 C= n (for any given nonzero integer n) satisfying the property that for any X∈ N , the period length ℓ k=ℓ( D k(X) ) of the simple continued fraction expansion of D k(X) is constant for fixed k and lim k→∞ ℓ k =∞. 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).