The irrationality exponent of π is 2

What OpenAI claims

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 |π-p/q| ≥ q^-2-ε. This also proves convergence of the Flint–Hills series ∑_{n ≥ 1}1/(n³ sin² n), with angles in radians. (Lean)

Paper

1. The irrationality exponent of pi is 2

   24 September 2026 · Preprint, not peer reviewed · PDF

   We prove the conjecture that the irrationality exponent of π is 2. As a consequence, the classical Flint–Hills series ∑_{n ≥ 1}1/(n³ sin² n) converges, with angles in radians.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.