Theon's Ladder for Any Root
IJMEST Volume 36, Number 4, ISSN 0020-739X
Theon's ladder is an ancient algorithm for calculating rational approximations for the square root of 2. It features two columns of integers (called a ladder), in which the ratio of the two numbers in each row is an approximation to the square root of 2. It is remarkable for its simplicity. This algorithm can easily be generalized to find rational approximations to any square root. In this paper we show how Theon's original method is naturally generalized for the calculation of any root, n square root of c, where 1 is less than c. In the generalization given here we require n columns of numbers as we generate rational approximations to an nth root. Several different recursion relations for the numbers that appear in the ladder are given, and a generating function for calculating the nth row of the ladder is found. Methods of increasing the rate of convergence are given, and a method of reducing the n-column ladder to a 2-column ladder is shown.
Osler, T.J., Wright, M. & Orchard, M. (2005). Theon's Ladder for Any Root. International Journal of Mathematical Education in Science and Technology, 36(4), 389-398.