The irrationality exponent of pi is 2

The irrationality exponent of an irrational $x$ is $\mu(x)=\sup\{\nu>0: 0<|x-p/q|<q^{-\nu}$ for infinitely many coprime $p,q$ , $q\ge2\}$ ; Dirichlet's pigeonhole argument gives $\mu(x)\ge2$ , and almost every real has $\mu=2$ . For $\pi$ only finite upper bounds were known: Mahler (1953) gave $42$ , Mignotte $20$ , Hata about $8.016$ , Salikhov about $7.606$ , and Zeilberger and Zudilin about $7.103$ . The expected value is $2$ , recorded as the conjecture that for every $\varepsilon>0$ , $|\pi-p/q|\ge q^{-2-\varepsilon}$ for all large $q$ . Is $\mu(\pi)=2$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Diophantine approximation
Posed by: Folklore conjecture; the manuscript cites its statement in Michel Waldschmidt, Open Diophantine problems, Moscow Math. J. 4 (2004), p. 265
Year posed: —
Years open: —
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 60 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: $\mu(\pi)=2$ ; for every $\nu>2$ there is $Q(\nu)$ with $|\pi-p/q|\ge q^{-\nu}$ for all $p\in\mathbb Z$ and $q\ge Q(\nu)$ . Corollary: the Flint Hills series converges, and $\sum n^{-a}|\sin n|^{-b}$ converges iff $a>\max\{1,b\}$ . The bound is NOT effective (no explicit $Q(\nu)$ ), and it is NOT a bound of the form $c/q^2$ : bounded partial quotients for $\pi$ are not claimed. The method is an interpolation-determinant argument, not an improvement of the integral constructions behind earlier numerical bounds.

What the AI did

The release README says the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result, out of roughly 4,000 problems posed; the output was aggregated into result families and manuscripts and kept if judged significant enough. This family has one manuscript, dated September 24, 2026. The manuscript is credited to 'OpenAI' alone and names no human author. The README's two exceptions to the fixed procedure (the Riemann zeta zero-free region work, whose Re(s) > 11/12 write-up was human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The README also cautions that unformalized results could have issues. OpenAI also released an abridged reasoning summary for this family (reasoning_traces/irrationality-exponent-of-pi.pdf).

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source read against the conjecture as posed; it gives, for every real $\nu>2$ , a threshold $Q(\nu)$ with $|\pi-p/q|\ge q^{-\nu}$ for all integers $p$ and $q\ge Q(\nu)$ , which is $\mu(\pi)=2$ . The threshold is ineffective. The proof was not refereed. Lean: the release's Comparator challenge PiExponent (OAI.PiExponent.main) with solution module OAI.NumberTheory.PiExponent.Main, both present at the pinned commit; the challenge is not in the formalization catalogue. Its statement was read here and covers the headline: the eventual lower bound for every $\nu>2$ and the supremum characterization over rationals equal to 2. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The manuscript notes Carella (2022) also claimed exponent two and points out a sign error in that paper's proof of a stronger claim.

The irrationality exponent of pi is 2

What was actually shown

What the AI did

Verification

Sources

Discussion