Lagrange proved the converse of Euler's theorem: if "x" is a quadratic irrational, then the regular continued fraction expansion of "x" is periodic. Given a quadratic irrational "x" one can construct "m" different quadratic equations, each with the same discriminant, that relate the successive complete quotients of the regular continued fraction expansion of "x" to one another. Since there are only finitely many of these equations (the coefficients are bounded), the complete quotients (and also the partial denominators) in the regular continued fraction that represents "x" must eventually repeat.