Result 017, Number theory

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.

Lean formalization Reasoning summary (PDF) Proof

The bigger picture

Why it matters

An unreviewed manuscript claims to pin down how closely fractions can approximate pi. The claimed limit also settles whether a particular infinite sum remains finite despite terms that can become unusually large.

What changes?

The irrationality exponent measures how strong a power-law approximation by fractions can occur infinitely often. The manuscript reports that pi's exponent is exactly 2. Specifically, for every positive epsilon, there is a threshold depending on epsilon such that, for every integer p and positive integer q beyond that threshold, the distance between pi and p/q is at least the reciprocal of q raised to the power (2 plus epsilon). This is an eventual bound, not one asserted for every denominator.

What does that help mathematicians do?

The manuscript reports convergence of the Flint-Hills series: the sum, over positive integers n, of the reciprocal of n cubed times sine squared of n, with angles in radians. Its difficulty comes from integers lying close to multiples of pi, where sine is small and the summand can spike. The claimed consequence says those spikes collectively cannot prevent the sum from having a finite value.

Are there practical applications?

The immediate value is foundational: the result would establish a precise limit on exceptionally accurate rational approximations to pi and resolve a convergence question governed by those approximations. The supplied material presents the Flint-Hills conclusion as a mathematical consequence, not a faster method for computing pi or a practical numerical algorithm.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The irrationality exponent of pi is 2

September 24, 2026 23 pages

We prove the conjecture that the irrationality exponent of π is 2. As a consequence, the classical Flint–Hills series ∑n≥11/(n3sin⁡2n)\sum_{n\ge1}1/(n^3\sin^2 n) converges, with angles in radians.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-irrationality-exponent-of-pi-is-2-September-24-2026,
  author = {{OpenAI}},
  title = {{The irrationality exponent of $\pi$ is $2$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf}{OAI:The-irrationality-exponent-of-pi-is-2-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/017.md.

The irrationality exponent of π is 2

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves that the irrationality exponent of π\pi is exactly two. For every ν>2\nu>2, all sufficiently large positive denominators qq satisfy ∣π−p/q∣≥q−ν|\pi-p/q|\ge q^{-\nu} for every integer numerator pp. It also states the exact supremum characterization using infinitely many rational approximations.

The paper's convergence consequence for the Flint–Hills series is outside this selected statement.

Comparator links

Result Comparator statement
The irrationality exponent of π\pi equals two PiExponent.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.