Remark: Family 017. The irrationality exponent of π is 2

Proves that the irrationality exponent of $\pi$ is exactly $2$: for every $\epsilon >0$ and all sufficiently large denominators $q$, every rational $p/q$ satisfies $|\pi -p/q|\ge q^{-2-\epsilon }$. This also proves convergence of the Flint–Hills series ${\sum }_{n\ge 1}1/(n^3{\sin }^{2}n)$, with angles in radians.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.