Unbounded continued-fraction coefficients exclude quadratic irrationality
= Unbounded continued-fraction coefficients exclude quadratic irrationality
For an irrational $x$ with <continued fraction convergents> $p_n/q_n$, one has $|x-p_n/q_n|\leq1/(q_nq_{n+1})\leq1/(a_{n+1}q_n^2)$. The <Liouville approximation theorem> for a <quadratic irrational> gives a positive lower bound $C/q_n^2$ instead. Thus unbounded partial quotients preclude quadratic irrationality, and in particular imply $x^2$ is irrational.