The irrationality measure of pi

How well can $\pi$ be approximated by fractions? Almost every real number has irrationality measure 2, and $\pi$ is expected to as well, but the best accepted upper bound is about 7.1019; each improvement comes from an explicit family of integrals that produce good rational approximations. A September 2026 preprint claims a proof that the measure is exactly 2; it has not yet been checked independently.

Problem

The irrationality measure $\mu(x)$ of a real number $x$ is the supremum of all $\mu$ such that

$$\left| x - \frac{p}{q} \right| < \frac{1}{q^{\mu}}$$

has infinitely many solutions in integers $p$ and $q \ge 1$. Equivalently, it is the infimum of the $\nu$ such that $|x - p/q| > q^{-\nu-\varepsilon}$ for every $\varepsilon > 0$ and all sufficiently large $q$.

Every irrational number has $\mu(x) \ge 2$ (Dirichlet). Determine $\mu(\pi)$, or lower the known upper bound $\mu(\pi) < 7.101862832357$.

Background

Mahler (1953) first proved that $\mu(\pi)$ is finite, with the bound 42, so $\pi$ is not a Liouville number. Later improvements follow one pattern: build small integer linear forms in $1$ and $\pi$, typically from explicit integrals or hypergeometric constructions, and convert their size and growth into a bound on $\mu(\pi)$. The method has been refined by Mignotte, Chudnovsky, Rhin and Viola, Hata, Salikhov, and Zeilberger and Zudilin, often with computer algebra guiding the choice of integrals. It is widely conjectured that $\mu(\pi) = 2$, as for almost every real number, but nothing better than 2 is known as a lower bound.

What is known

• Upper bounds: $42$ (Mahler, 1953); $20.6$ (Mignotte, 1974); $19.8899944$ (Chudnovsky, 1982); $14.797074$ (Rhin and Viola, 1993, from a measure for $\zeta(2) = \pi^2/6$); $8.016045$ (Hata, 1993); $7.606308$ (Salikhov, 2008); $7.103205334137\ldots$ (Zeilberger and Zudilin, 2020).
• Claimed resolution: an OpenAI preprint, The irrationality exponent of π is 2 (24 September 2026), claims $\mu(\pi) = 2$ and, as a corollary, convergence of the Flint–Hills series $\sum 1/(n^3 \sin^2 n)$. It is not refereed or formalized, and the source does not list it.
• Best accepted bound: $\mu(\pi) < 7.101862832357$ (Bai, September 2026 preprint), adding two independent numerator exponents to the Zeilberger–Zudilin integral and choosing them equal to $1857/2785$. Bai notes that this parameter point is a local minimizer of the bound, not necessarily a global one.
• Lower bound: $2$, which holds for every irrational number.

What would count as progress

• An independent check of the claimed proof that $\mu(\pi) = 2$: confirmation settles the problem, and a gap reopens it.
• A new family of integrals or linear forms giving $\mu(\pi) < 7.10186$, with a proof. Its asymptotic and arithmetic estimates can be checked with computer algebra.
• An exploration of the Bai–Zeilberger–Zudilin parameter space that finds a better admissible point, with the bound evaluated rigorously.

References

1. M. Hata, "Rational approximations to π and some other numbers", Acta Arithmetica 63 (1993), 335–349. doi:10.4064/aa-63-4-335-349
2. V. Kh. Salikhov, "On the irrationality measure of π", Russian Mathematical Surveys 63 (2008), 570–572. doi:10.1070/RM2008v063n03ABEH004543
3. D. Zeilberger and W. Zudilin, "The irrationality measure of π is at most 7.103205334137…", Moscow Journal of Combinatorics and Number Theory 9 (2020), 407–419. doi:10.2140/moscow.2020.9.407, arXiv:1912.06345
4. Y. Bai, "The irrationality measure of π is at most 7.101862832357" (2026). arXiv:2609.11276

Source

Imported from Optimization Constants in Mathematics, constant 7a, curated by Terence Tao and contributors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.