On Elementary Properties of a Multi-Dimensional Generalization of the Euclidean Algorithm (PDF)
In 1848, Charles Hermite posed the question of whether every real number admits a representation as a sequence of integers, periodic if and only if the number is a cubic irrational. While continued fractions solve this problem for quadratic irrationals, no analogous representation is known for higher-degree algebraic numbers. This thesis explores a new approach to Hermite’s question through the construction of multidimensional continued fractions (MDCFs), which extend ordinary continued fractions to higher dimensions using a generalization of the Euclidean algorithm.
It is shown that most multidimensional continued fractions converge, and that any periodic fraction represents an algebraic number of bounded degree. These results establish one direction of Hermite’s question in higher dimensions. An experimental study complements the theoretical findings, providing numerical evidence that a wide class of algebraic numbers -- including many cube roots -- give rise to periodic MDCFs.