OpenAI's π claim: how it tries to reach the exact exponent

On this page

π is familiar enough to be taught in school, yet some of its basic properties still require original research. An exact limit on how well fractions can approximate it would describe infinitely many cases no computation could inspect. OpenAI now attributes a proposed proof of that limit to an internal model. We think the significance includes what such systems could let people investigate, especially when a question's value is knowledge itself.

The particular question has a long history of human effort. Kurt Mahler put a finite upper bound on π's irrationality measure in 1953. Over the next seventy-three years, mathematicians brought that bound down from 42 to a little above 7. The expected exact answer remained 2. OpenAI's 6 October mathematics release includes a manuscript claiming to reach it.

We followed the older methods and OpenAI's abridged reasoning summary to understand how it attempts that jump. The proposed proof changes the auxiliary polynomials used to turn exceptionally good fractions into a contradiction. Its failed constructions and repairs let us examine the inventive work behind the claim. They also make the wider question concrete. Who will be able to use this kind of research capability to pursue questions of their own?

A brilliant fraction leaves the hard question open

We calculated the errors of two familiar fractions using Python's standard double-precision π. These rounded numbers illustrate the question; they aren't certified bounds.

Fraction	Approximate absolute error	Error multiplied by q²
22/7	0.00126449	0.0619600
355/113	0.000000266764	0.00340631

355/113 is the better approximation even after allowing for its larger denominator. Nothing about that one fraction, however, tells us how often exceptionally close approximations keep appearing.

The irrationality exponent captures that question. It is the supremum of the exponents ν for which an irrational number x has infinitely many reduced fractions $p/q$ satisfying $0<|x-p/q|<q^{-\nu}$. Every irrational real number has exponent at least 2. Zeilberger and Zudilin explain the equivalent upper-bound formulation used here.

OpenAI claims the matching upper bound for π. For every positive $\varepsilon$, there would be a threshold $Q(\varepsilon)$ beyond which every integer numerator p and denominator q satisfies:

$$\left|\pi-\frac{p}{q}\right|\ge\frac{1}{q^{2+\varepsilon}}\qquad(q\ge Q(\varepsilon)).$$

The threshold can depend on $\varepsilon$. The paper gives no effective threshold. It also doesn't establish one positive constant $c$ with $|\pi-p/q|\ge c/q^2$ for all denominators; that stronger property would require bounded continued-fraction partial quotients.

That is why calculating more digits can't settle this claim. The theorem has to govern all sufficiently large denominators, including the ones nobody has inspected. A spectacular fraction is compatible with exponent 2. An endless supply beating a fixed exponent greater than 2 would not be.

Why we think the jump matters

Following the history led us back to Kurt Mahler—the same mathematician behind the convex-geometry problem elsewhere in this release. In 1953 he established the finite upper bound 42 for π's irrationality measure. Knowing that no fraction equals π leaves open how exceptionally close fractions can keep getting. A finite bound begins to control that second question.

Zeilberger and Zudilin's account traces subsequent improvements through Maurice Mignotte, Gregory Chudnovsky and Masayoshi Hata to Vladislav Salikhov. Salikhov announced a bound of about 7.606308 in 2008 and published it in 2010. The history is a succession of stronger limits, rather than a procession of increasingly long decimal expansions of π.

Zeilberger and Zudilin's 2019 work established an upper bound of approximately 7.103205334137. Their method varies a complex contour integral used by Salikhov. The integral produces expressions of the form $A_n+B_n\pi$ with rational coefficients. The aim is to make those expressions small while keeping the arithmetic cost of their coefficients under control.

The two demands compete. Clearing the coefficients' denominators makes an integer comparison possible, but can enlarge the expression too much. Better divisibility and faster analytic decay improve the resulting irrationality bound. The authors used Maple and recurrence calculations to locate promising integral parameters, then proved the necessary arithmetic savings and asymptotic estimates rigorously. This was already computer-assisted discovery, with a human-readable proof; it was not merely a numerical search for good fractions.

A September 2026 preprint by Yufei Bai claims an improvement to below 7.101862832357. That remains a preprint claim. Neither bound says the true exponent is around 7; each limits how large it could be.

OpenAI's manuscript proposes a different interpolation statement and cost balance, rather than another parameter choice for that integral. The question is how it tries to make the arithmetic and analytic estimates compatible for every fixed exponent above 2.

The construction has to survive its own choices

The proposed proof begins by assuming that fractions approximate π too well, with some fixed exponent $\nu>2$ at arbitrarily large denominators. It selects several such fractions with widely separated denominators and tries to turn them into a contradiction between arithmetic and analysis.

A determinant is a number calculated from a square array. If the entries are integers and the determinant is nonzero, its absolute value is at least 1. The paper works with more complicated entries whose denominators must first be controlled, but the principle is the same: arithmetic gives a floor below which a nonzero quantity cannot fall.

The analytic side tries to push the same quantity below that floor. The supposed exceptional approximations provide very small errors. The estimates divide into two cases. In one case, high powers of the tiny approximation errors make the determinant terms small. The other case handles terms where those powers do not provide enough saving: repeated degrees in the Taylor expansions make some determinant columns proportional, so those terms vanish. A Taylor expansion expresses a function near a point as successive powers. Surviving terms must use sufficiently high distinct degrees to provide the required saving. A carefully chosen polynomial family has to make either route produce enough saving. Small enough on one side, nonzero and too large on the other: that would rule out the assumed infinite supply of exceptional fractions. The manuscript's introduction and determinant comparison set out this architecture.

The proposed change lies in the polynomial family and its interpolation theorem. Theorem 2.1 asks weighted polynomials to realise specified Taylor data at several points on logarithmic curves. Taylor data means the values of a function and selected derivatives near a point. The proof needs enough independent control over those values to arrange the cancellation without forcing the determinant to vanish.

Counting coefficients cannot establish that control. A set of conditions can contain hidden dependencies, just as two equations can repeat the same information. The interpolation map must have full rank. The weights also have to be fixed before point locations that later depend on the exceptional fractions. Reversing that order could make the contradiction circular.

Interpolation and determinants already belong to the human history of this problem; the manuscript itself describes Mahler's use of logarithmic combinations and determinant nonvanishing. The claimed step is a multiple-centre interpolation statement with separated weights, together with a cost balance strong enough to make the contradiction work for every fixed $\nu>2$. Its choices must remain independent in the order the proof requires. That is where the new argument has to improve on the older tradeoff.

What the released reasoning tells us about the search

The abridged reasoning summary gives a more informative picture than the finished theorem. Its first part pursues the weaker bound 62/25, enough for the Flint–Hills application. Part II, beginning at §45, makes a separate attempt at exponent 2 using an interpolation-and-determinant technique; it does not assume that the earlier 62/25 claim has been proved.

The account records attempts using rational approximants and logarithmic or exponential constructions. Several routes lose too much as coefficients or denominators grow; others fail to keep enough independent conditions or establish that the required determinant is nonzero. These are versions of the arithmetic-versus-smallness problem visible in the older integral methods.

The later construction changes the allowed polynomial degree pattern, separates denominator scales and seeks independent Taylor conditions through a geometric interpolation argument. The reported search also draws inspiration from an earlier OpenAI Catalan determinant paper. That is a more specific account of the shift: change the auxiliary construction and its independence guarantees so the small-error estimates are not consumed by their arithmetic costs.

The summary lets us follow the proposed redesign, but not the full provenance of every step. It is OpenAI's abridged account; it does not disclose the exact original question or every per-attempt human intervention. The finished proof still has to establish the interpolation theorem and both sides of the determinant comparison.

The strange spike at 355

There is a wonderfully concrete consequence hiding behind the exponent. The Flint–Hills series is

$$\sum_{n=1}^{\infty}\frac{1}{n^3\sin^2 n}.$$

We sum over positive integers $n$, with angles in radians. When an integer is close to a multiple of π, its sine becomes tiny and the term can become enormous compared with its neighbours.

Our calculation puts the term at n = 355 at about 24.5982. At 354 and 356, the terms are about 0.0000000318 and 0.0000000313. The fraction 355/113 explains the spike.

That makes a finite plot a poor way to decide whether the infinite series converges. A smooth stretch of small terms can be interrupted by an exceptional approximation.

Alekseyev showed that convergence requires π's exponent to be at most 5/2. Meiburg proved that an exponent strictly below 5/2 is sufficient. The equality case remains a separate issue. OpenAI's claimed 2 would meet the strict condition.

Even here, the explanation needs care. A crude bound on each term can produce a divergent upper-bound series; that doesn't prove Flint–Hills diverges. The missing ingredient is how frequently large terms can occur. The manuscript's spacing argument uses differences between denominators to limit how densely those spikes accumulate.

That is what makes the claim compelling to us. A universal statement about fractions could answer a question about an infinite series whose dangerous terms are rare and easy to miss.

Who gets to pursue the next question?

An exponent-2 proof would settle a precise question about π and give number theorists a new argument to examine. It would also add to a history in which people and computers already worked together. Zeilberger and Zudilin used computers to discover better integral parameters and supplied a human-readable proof of their main bound. Their account includes rigorous computer-generated proofs of coarser bounds. The proposed advance here is this construction and exact result, rather than the arrival of machines in mathematics.

What interests us is the possibility of more people being able to pursue difficult fundamental questions. The released search describes the invention of an auxiliary object, failures in its cost balance and attempts to repair it. If reliable systems can help with that work, a researcher could explore approaches that would otherwise demand more time or specialist collaboration than they can obtain. How widely that opportunity spreads will depend on access to the tools and the resources needed to use them.

The independent mathematics advisory group's response to this release makes that issue explicit. It argues that mathematicians need the freedom and resources to formulate their own questions, beyond understanding results selected by AI labs. Its advisory role is not an endorsement of the results. We share the concern about who gets to set the research agenda.

If the strongest systems remain confined to a few companies, those companies could gain unusual influence over which problems receive sustained computational effort. Publishing a finished manuscript would give the community a result to assess. It would not give a student or an independent researcher the same opportunity to develop an idea of their own. The distinction matters across science, where the questions a community can afford to ask help determine what it discovers.

We want agents to expand curiosity-driven research, including work whose eventual use nobody can yet specify. A valid proof about π would be a meaningful contribution to knowledge. The larger opportunity would be to give more researchers the means to attempt the next fundamental question, rather than make a small number of laboratories the places everyone else waits for answers.

Source note: we read selected statements and proof sections in the manuscript, compared the literature linked above and calculated the illustrative fractions and series terms with Python. We also read the relevant released reasoning summary where cited. Sources checked 8 October 2026. We haven't independently certified the proof or run a Lean build.