Naslov
Newton's formula and continued fraction expansion of sqrt(d)

Autori
Dujella, Andrej

Izvornik
Experimental Mathematics (1058-6458) 10 (2001), 1; 125-131

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
continued fractions; Newton's formula

Sažetak
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 we have R_n = p_{2n+1} / q_{2n+1} for all n >= 0. Motivated with this fact we define two numbers j = j(d,n) and b=b(d) by R_n = p_{2n+1+2j} / q_{2n+1+2j} if R_n is a convergent of sqrt(d); b = | {n : 0 <= n <= s-1 and R_n is a convergent of sqrt(d)} |. The question is how large the quantities |j| and b can be. We prove that |j| is unbounded and give some examples which support a conjecture that b is unbounded too. We also discuss the magnitude of |j| and b compared with d and s(d).

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA