Infinite Families of Pellian Polynomials and Their Continued Fraction Expansions

Abstract

We investigate families \( \lbrace D_k(X)\rbrace_{k\in{\rm N}} \) of quadratic integral polynomials and show that, for a fixed k ∈ N and arbitrary X ∈ N, the period length of the simple continued fraction expansion of \( \sqrt {D_k(X)} \) is constant. Furthermore, we show that the period lengths of \( \sqrt {D_k(X)} \) go to infinity with k. For each member of the families involved, we show how to easily determine the fundamental unit of the underlying quadratic field. We also demonstrate how the simple continued fraction expansion of \( \sqrt {D_k(X)} \) is related to that of \( \sqrt {C} \). This continues work in [3]-[5].