that the th root of the denominator of the th convergent of the continued fraction expansion of almost all real numbers approaches an asymptotic limit, approximately 3.27582, which is known as Lévy's constant. Lochs' theorem states that th convergent of the continued fraction expansion of almost all real numbers determines the number to an average accuracy of just over decimal places. Applications Square roots Generalized continued fractions are used in a method for computing square roots. The identity leads via recursion to the generalized continued fraction for any square root: Pell's equation Continued fractions play an essential role in the solution of Pell's equation. For example, for positive integers and , and non-square , it is true that if , then is a convergent of the regular continued fraction for . The converse holds if the period of the regular continued fraction for is 1, and in general the period describes which convergents give solutions to Pell's equation. Dynamical systems Continued fractions also play a role in the study of dynamical systems, where they tie together the Farey fractions which are seen in the Mandelbrot set with Minkowski's question mark function and the modular group Gamma. The backwards shift operator for continued fractions is the map called the Gauss map, which lops off digits of a continued fraction expansion: . The transfer operator of this map is called the Gauss–Kuzmin–Wirsing operator. The distribution of the digits in continued fractions is given by the zero'th eigenvector of this operator, and is called the Gauss–Kuzmin distribution. Eigenvalues and eigenvectors The Lanczos algorithm uses a continued fraction expansion to iteratively approximate the eigenvalues and eigenvectors of a large sparse matrix. Networking applications Continued fractions have also been used in modelling optimization problems for wireless network virtualization to find a route between a source and a destination. Examples of rational and irrational numbers ra: rational approximant obtained by expanding continued fraction up to ar History 300 BCE Euclid's Elements contains an algorithm for the greatest common divisor, whose modern version generates a continued fraction as the sequence of quotients of successive Euclidean divisions that occur in it. 499 The Aryabhatiya contains the solution of indeterminate equations using continued fractions 1572 Rafael Bombelli, L'Algebra Opera – method for the extraction of square roots which is related to continued fractions 1613 Pietro Cataldi, Trattato del modo brevissimo di trovar la radice quadra delli numeri – first