A short proof that the number of division steps in the Euclidean algorithm is normally distributed
Year: 2015
Type: preprint
Abstract: D. Hensley showed in 1994 that the number of steps taken by the Euclidean algorithm to find the greatest common divisor of two natural numbers less than or equal to n follows a normal distribution in the limit as n tends to infinity. V. Baladi and B. Vallée subsequently gave an alternative proof for both the classical Euclidean algorithm and several of its close variants, based on a detailed investigation of spectral properties of the transfer operator associated to the Gauss map, building on deep results of D. Dolgopyat. In this article we give a much shorter, albeit less quantitative, proof of this result using only basic spectral properties of the transfer operator together with the method of moments and a Tauberian theorem due to H. Delange.
Source: arXiv (Cornell University)
Author: Morris, Ian D.
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G2578688790
OpenAlex ID: W1567630643