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Newton's approximants and continued fractions

Let d be a positive integer which is not a perfect square. Let p_n/q_n denote the n-th convergent and s(d) the length of the shortest period in the simple continued fraction expansion of sqrt(d). It was proved by Mikusinski in 1954 that if 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 nonegative integers n. If r_n is a convergent of sqrt(d), then we say that R_n is a "good approximant". In 2001, we proved the converse of Mikusinski's result, namely that if all approximants are good, then s(d)<=2. It is easy to see that R_n>sqrt(d). Therefore, good approximants satisfy R_n=p{; ; 2n+1+2j}; ; /q_{; ; 2n+1+2j}; ; for an integer j=j(d, n). If s(d)<=2, then j(d, n)=0. For s(d)>2, we proved the upper bound |j(d, n)|<=(s(d)-3)/2, and we presented a sequence of d's (given in terms of Fibonacci numbers) which shows that this upper bound for |j(d, n)| is sharp. Let b(d) denote the number of good approximants among the numbers R_n, n=0, 1, ..., s(d)-1. We will present some results and conjectures (based on experimental data) about the magnitude of s(d) compared with d and s(d).

continued fractions; Newton's formula

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