Newton's Formula and the Continued Fraction Expansion of √d

Abstract

It is known that if the period s(d) of the continued fraction expansion of √d satisfies s(d) ≤ 2, then all Newton's approximants

are convergents of √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 Rn is a convergent of √d, and define b = b(d) by b = |{n : 0 ≤ n ≤ s-1 and R_n is a convergent of √d}|. The question is how large |j| and b can be. We prove that |j| is unbounded and give some examples supporting a conjecture that b is unbounded too. We also discuss the magnitude of |j| and b compared with d and s(d),

Keywords:

• continued fractions
• Newton's formula