Ustinov, A. V. The mean number of steps in the Euclidean algorithm with least absolute value remainders. (English. Russian original) Zbl 1208.11014 Math. Notes 85, No. 1, 142-145 (2009); translation from Mat. Zametki 85, No. 1, 153-156 (2009). It is well known that the Euclidean algorithm in which the remainders \[ a=bq+r, \qquad q=\left[\frac{a}{b}+\frac{1}{2} \right], \qquad -\frac{q}{2} \leq r < \frac{q}{2} \] with least absolute values are chosen leads to the decomposition in a continued fraction \[ \frac{a}{b}=t_0+\cfrac{{\varepsilon_1}}{ t_1+\cfrac{{\varepsilon_2}}{ t_2+ \ddots + \cfrac{{\varepsilon_l}}{{t_l}}}} \] of length \(l=l(a/b)\), where \(t_0\) is an integer, \(t_1,t_2,\dots,t_l\) are positive integers, and \[ \varepsilon_k=\pm 1, t_k \geq 2, k=1,\dots,l, \; t_k+\varepsilon_{k+1}\geq 2, k=1,\dots,l-1. \] Two estimations for the mean number of steps in the Euclidean algorithm with the choice of least absolute value remainders are obtained in the paper.The bibliography contains 5 sources. Reviewer: Michael M. Pahirya (Mukachevo) Cited in 4 Documents MSC: 11A55 Continued fractions 11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors 11J70 Continued fractions and generalizations 11Y16 Number-theoretic algorithms; complexity Keywords:Euclidean algorithm; Euclidean algorithm with least absolute value remainder; continued fraction; Gauss–Kuzmin statistics PDFBibTeX XMLCite \textit{A. V. Ustinov}, Math. Notes 85, No. 1, 142--145 (2009; Zbl 1208.11014); translation from Mat. Zametki 85, No. 1, 153--156 (2009) Full Text: DOI References: [1] O. Perron, Die Lehre von den Kettenbrüchen, Vol. I: Elementare Kettenbrüche (B. G. Teubner, Stuttgart, 1954). · Zbl 0056.05901 [2] J.W. Porter, Mathematika 22(1), 20 (1975). · Zbl 0316.10019 [3] A. V. Ustinov, Izv. Izv. Ross. Akad. Nauk Ser. Mat. Ser. Mat. 72(5), 189 (2008) [Russian Acad. Sci. Izv. Math. 72 (5), 1023 (2008)]. [4] V. Baladi and B. Vallée, J. Number Theory 110(2), 331 (2005). · Zbl 1114.11092 [5] A. V. Ustinov, Algebra Anal. 20(5), 186 (2008). под графиком This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.