Integer norm lower bound for rational approximation to a quadratic irrational ($|\sqrt{D}-a/q|\ge c_D/q^2$)

For a positive nonsquare integer $D$ and positive integers $a,q$, the integer $D{q}^2-a^2$ is nonzero. Therefore $|\sqrt{D}-a/q|=|D{q}^2-a^2|/(q^2(\sqrt{D}+a/q))$. If $a/q\le \sqrt{D}+1$, the denominator factor is at most $2\sqrt{D}+1$; otherwise the approximation error already exceeds one. This proves that a quadratic irrational of this form is a badly approximable number. For $D=2$, splitting instead at $a/q=2$ gives the strict bound $|\sqrt{2}-a/q|>1/(4q^2)$.

// Target: analytic-number-theory.bigb