NEWTON’S FORMULA AND THE CONTINUED FRACTION EXPANSION OF \sqrt{d}

ANDREJ DUJELLA

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} = \frac{1}{2}(\frac{p_{n}}{q_{n}} + \frac{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}, and define b = b(d) by b = |\{n : 0 ≤ n ≤ s−1 and 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 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).