Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational p/q satisfies |\pi-p/q|\ge q^{-2-\varepsilon}. This also proves convergence of the Flint–Hills series \sum_{n\ge1}1/(n^3\sin^2 n), with angles in radians. (Lean)
Manuscript
The irrationality exponent of pi is 2
September 24, 2026 · source · cite as \cite{OAI:The-irrationality-exponent-of-pi-is-2-September-24-2026}
We prove the conjecture that the irrationality exponent of π is 2. As a consequence, the classical Flint–Hills series \sum_{n\ge1}1/(n^3\sin^2 n) converges, with angles in radians.
Result 017 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md