On the period length of some special continued fractions. (English) Zbl 0766.11003

For positive integers \(r\), \(s\) with \(s>r\) let \(M(r,s)\) be the length of the simple continued fraction expansion of \(s/r\). Let \(gcd(a,q)=1\) where \(a\) and \(q\) are positive integers with \(a=1\).
Define natural numbers \(s_ i\) by \(s_ i\equiv a^ i\pmod q\) and \(0<s_ i<q\). Further let \(W(a,q)=2\sum_{i=1}^ \omega \lfloor(M(s_ i,q)+1)/2\rfloor\), where \(\omega\) is the order of \(a\) modulo \(q\), and \(\lfloor\;\rfloor\) denotes the greatest integer function. If \(q=a^ m\pm 1\), then, it is easy to evaluate \(W(a,q)\), e.g. \(W*a,a^ m+1)=2m-2\). In the paper the authors show how to develop formulas for determining the value of \(W(a,(a^ m\pm1)/q)\) in terms of \(W(a,q)\) when \(a^ m\equiv\pm 1\pmod q\).
Reviewer: Péter Kiss (Eger)


11A55 Continued fractions
11J70 Continued fractions and generalizations
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[1] Bernstein, L., Fundamental units and cycles, J. Number Theory8 (1976), 446-491. · Zbl 0352.10002
[2] Knuth, D.E., The Art of Computer Programing II: Seminumerical Algorithms, Addison-Wesley, 1981.
[3] Mollin, R.A. and Williams, H.C., Consecutive powers in continued fractions, (to appear: Acta Arithmetica). · Zbl 0764.11010
[4] Perron, O., Die Lehre von den Kettenbrüchen, Chelsea, New-York (undated). · JFM 43.0283.04
[5] Porter, J.W., On a theorem of Heilbronn, Mathematika22 (1975), 20-28. · Zbl 0316.10019
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