  The theory of continued fractions has been generalized to l-adic numbers by
several authors and presents many differences with respect to the real case. In
the present paper we investigate the expansion of rationals and quadratic
irrationals for the l-adic continued fractions introduced by Ruban. In this
case, rational numbers may have a periodic non-terminating continued fraction
expansion, moreover, for quadratic irrational numbers, no analogue of
Lagrange's theorem holds. We give general explicit criteria to establish the
periodicity of the expansion in both the rational and the quadratic case (for
rationals, the qualitative result is due to Laohakosol).

Number Theory