An Algorithm for Constructing Multidimensional Continued Fractions and Linear Dependence of Numbers

The Güting algorithm for constructing multidimensional continued fractions is considered. It is proved that, in the case of dimension $2$, this algorithm can be used to find the coefficients of the linear dependence of numbers; a criterion is given for verifying that the partial quotients furnished by the algorithm are, indeed, elements of the continued fraction for the expanded (generally irrational) numbers.