engleski

Square roots with many good approximants

Let d be a positive integer that is not a perfect square. It was proved by Mikusinski in 1954 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). If R_n is a convergent of sqrt(d), then we say that R_n is a good approximant. Let b(d) denote the number of good approximants among the numbers R_n, n=0, 1, ..., s(d)-1. In this paper we show that the quantity b(d) can be arbitrary large. Moreover, we construct families of examples which show that for every positive integer b there exist a positive integer d such that b(d)=b and b(d) > s(d)/2.

continued fractions; Newton's method

Rad je prezentiran na skupu The 2004 Number Theoretic Algorithms and Related Topics Workshop ; AO6.