## Experimental Mathematics

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

### 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).

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 30 August 2001

**Permanent link to this document**

https://projecteuclid.org/euclid.em/999188427

**Mathematical Reviews number (MathSciNet)**

MR1 822 858

**Zentralblatt MATH identifier**

0989.11003

**Subjects**

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

**Keywords**

continued fractions Newton's formula

#### Citation

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