We consider the problem of representing an arbitrary algebraic number by a periodic branching continuous fraction. Previous attempts of such representations produced a purely formal result and the fractions obtained actually converged to a different number. The present work solves this problem: we demonstrate that any positive algebraic number can be represented as a branching continuous fraction with positive integer elements. We suggest an algorithm for constructing such a continuous fraction that allows to compute it explicitely, given an integer coefficient polynomial one of whose roots is the algebraic number in question. This result can be naturally generalized to arbitrary algebraic numbers since any of them may be represented as a sum of an integer and a positive algebraic number.

number theory, periodic branching continuous fraction, algebraic number representation

Mathematical problems and theory of numerical methods