Infinite families of Pellian plynomials and their continued fraction expansions (Q1411449)

\textit{A. Schinzel} [Acta Arith. 6, 393--413 (1961; Zbl 0099.04003), ibid. 7, 287--298 (1962; Zbl 0112.28001), Corr. ibid. 47, 295 (1986; Zbl 0603.10006)] showed some forty years ago that a quadratic polynomial \(D(X)=A^2X^2+2BX+C\) taking integer values at integers \(N\) has the property that the continued fraction expansion of \(\sqrt{D(N)}\) has only finitely many different period lengths as \(N\) varies if and only if \((B^2-A^2C)\bigm | 4\gcd(A^2,B)^2\). Here and elsewhere, the author has made the useful observation that if \((x,y)=(B,A)\) is the minimal solution of \(x^2-Cy^2=1\) in positive integers, and if for \(k=1\), \(2\), \(\ldots\,\), one sets \(B_k+A_k\sqrt C:(B_k+A\sqrt C)^k\), then the polynomials \(D_k(X)=A_k^2X^2+2B_kX+C_k\) satisfy Schinzel's condition and for almost all \(N\) the numbers \(\sqrt{D_k(N)}\) have period lengths increasing linearly with \(k\). Moreover, one may then readily explicitly state the fundamental unit of the order \(\mathbb Z[\sqrt{D_k(N)}\,]\). Illustrative numerical examples are provided.