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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/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.

Manuscripts in this family · 1

Sep 24, 2026·paper.pdf

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.

Full abstract

OpenAI also released the model's reasoning trace for this result: reasoning trace (PDF).