Number theory · Oct 6, 2026

The irrationality exponent of π is 2

Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational p/qp/q satisfies ∣π−p/q∣≥q−2−ε|\pi-p/q|\ge q^{-2-\varepsilon}. This also proves convergence of the Flint–Hills series ∑n≥11/(n3sin⁡2n)\sum_{n\ge1}1/(n^3\sin^2 n), with angles in radians.

Contributors

Unreleased OpenAI model

AI systems: Unreleased OpenAI model

Sources