Newton's Formula and the Continued Fraction Expansion of $\sqrt{d}$

Abstract

It is known that if the period s(d) of the continued fraction expansion of $\sqrt{d}$ satisfies s(d) ≤ 2, then all Newton's approximants R_n = (1/2)((p_n/q_n) + (dq_n/p_n)) are convergents of $\sqrt{d}$, and moreover R_n = p_2n+1/q_2n+1 for all N ≤ 0. Motivated by this fact we define j = j(d, n) by R_n = p_2n+1+2J/q_2n+1+2j if R_n is a convergent of $\sqrt{d}$}|. The question is how large |j| and b can be. We prove that |j| is unbounded and gie some examples supporting a conjecture that b is unbounded too. We also discuss the magnitude of |j| and be compared with d and s(d).