Lagrange's continued fraction theorem states that the regular continued fraction of a real number is eventually periodic iff the number is a quadratic surd. Equivalently, it is an irrational number satisfying a quadratic equation with integer coefficients. Lagrange proved the theorem in 1770 (Olds 1963, p. 89).
Lagrange's Continued Fraction Theorem
See also
Periodic Continued Fraction, Quadratic Surd, Regular Continued FractionExplore with Wolfram|Alpha
References
Olds, C. D. Continued Fractions. New York: Random House, p. 89, 1963.Referenced on Wolfram|Alpha
Lagrange's Continued Fraction TheoremCite this as:
Weisstein, Eric W. "Lagrange's Continued Fraction Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LagrangesContinuedFractionTheorem.html