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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.

MSC:

11A55 Continued fractions
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
11J70 Continued fractions and generalizations
11Y16 Number-theoretic algorithms; complexity
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References:

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[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
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