Experimental Mathematics

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

Andrej Dujella


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 Rn = (1/2)((pn/qn) + (dqn/pn)) are convergents of $\sqrt{d}$, and moreover Rn = p2n+1/q2n+1 for all N ≤ 0. Motivated by this fact we define j = j(d, n) by Rn = p2n+1+2J/q2n+1+2j if Rn 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).

Article information

Experiment. Math., Volume 10, Issue 1 (2001), 125-131.

First available in Project Euclid: 30 August 2001

Permanent link to this document

Mathematical Reviews number (MathSciNet)
MR1 822 858

Zentralblatt MATH identifier

Primary: 11Axx: Elementary number theory {For analogues in number fields, see 11R04}

continued fractions Newton's formula


Dujella, Andrej. Newton's Formula and the Continued Fraction Expansion of $\sqrt{d}$. Experiment. Math. 10 (2001), no. 1, 125--131. https://projecteuclid.org/euclid.em/999188427

Export citation