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Original scientific paper

Newton's approximants and continued fraction expansion of 1+d2

Vinko Petričević ; Department of Mathematics, University of Zagreb, Zagreb, Croatia


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Abstract

Let d be a positive integer such that d1(mod4) and d is not a perfect square. It is well known that the continued fraction expansion of 1+d2 is periodic and symmetric, and if it has the period length 2, then all Newton's approximants Rn=pn2+d14qn2qn(2pnqn) are convergents of 1+d2 and then it holds Rn=p2n+1q2n+1 for all n0. We say that Rn is a good approximant if Rn is a convergent of 1+d2. When >2, then there is a good approximant in the half and at the end of the period.
In this paper we prove that being a good approximant is a palindromic and a periodic property. We show that when >2, there are Rn's, which are not good approximants. Further, we define the numbers j=j(d,n) by Rn=p2n+1+2jq2n+1+2j if Rn is a good approximant; and b=b(d)=|{n:0n1 and Rn is a good approximant}|. We construct sequences which show that |j| and b are unbounded.

Keywords

continued fractions; Newton's formula

Hrčak ID:

93236

URI

https://hrcak.srce.hr/93236

Publication date:

5.12.2012.

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