OpenAI research collection · 1 manuscript · Source checked 2026-10-19
Fractions can get very close to pi, but how close depends on their denominator. This manuscript claims to pin down the best possible long-run rate of approximation, settling the irrationality exponent of pi at two.
A familiar constant becomes a demanding test of new reasoning. The result concerns approximation limits; it does not make pi rational or change its digits.
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 \sum_{n\ge1}1/(n^3sin^2 n), with angles in radians.
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
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^3sin^2 n) converges, with angles in radians.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.
Explore the AGI Clock ↗