A short proof that the number of division steps in the Euclidean algorithm is normally distributed

TL;DR

This paper provides a concise proof that the number of division steps in the Euclidean algorithm follows a normal distribution as the input size grows large, using basic spectral analysis and classical probabilistic methods.

Contribution

It offers a shorter, less quantitative proof of the normal distribution of Euclidean algorithm steps, simplifying previous complex spectral approaches.

Findings

Number of steps is normally distributed as n increases.

Proof relies on basic spectral properties and classical probabilistic methods.

Simplifies previous proofs by using elementary spectral analysis and Tauberian theorems.

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. Vall\'ee 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.