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  SQUARE ROOTS WITH MANY GOOD APPROXIMANTS

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by Andrej Dujella, Vinko Petričević
http://lnr.math.hr/~duje/papers/dujevinko.pdf
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Abstract:

Let d be a positive integer that is not a perfect square. It was proved by Mikusiński in 1954 that if the period s(d) of the continued fraction expansion of √ d satisfies s(d) ≤ 2, then all Newton’s approximants Rn = 1 pn dqn ( + 2 qn pn) are convergents of √ d. If Rn is a convergent of √ d, then we say that Rn is a good approximant. Let b(d) denote the number of good approximants among the numbers Rn, 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. 1

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