Complete Quotients

Periodic Continued Fraction - Relation To Quadratic Irrationals
... By considering the complete quotients of periodic continued fractions, Euler was able to prove that if x is a regular periodic continued fraction, then x is a quadratic irrational number ... discriminant, that relate the successive complete quotients of the regular continued fraction expansion of x to one another ... 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 ...