= Integer norm lower bound for rational approximation to a quadratic irrational
{title2=$|\sqrt D-a/q|\geq c_D/q^2$}
For a positive nonsquare integer $D$ and positive integers $a,q$, the integer $Dq^2-a^2$ is nonzero. Therefore $|\sqrt D-a/q|=|Dq^2-a^2|/(q^2(\sqrt D+a/q))$. If $a/q\leq\sqrt D+1$, 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 $|\sqrt2-a/q|>1/(4q^2)$.
// Target: analytic-number-theory.bigb
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