The irrationality exponent of π is 2

1 · The essentials

How closely can a fraction approach $π$ ? The main result places a limit on this accuracy when the denominator becomes large. For every positive number $ε$ , however small, the distance between $π$ and a fraction $p / q$ is at least $q^{- 2 - ε}$ once $q$ is sufficiently large. The threshold can depend on $ε$ . The manuscript presents this as resolving a conjecture: the irrationality exponent of $π$ is exactly 2. It also derives that an infinite sum called the Flint–Hills series has a finite value. The main result has a Lean formalization, but the supplied information does not specify whether that formalization also covers this consequence.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine fractions as markers used to aim at $π$ . The result imposes a minimum distance between each marker and the target, depending on its denominator, once that denominator is sufficiently large; it does not say that every marker actually approaches this limit.

Why it matters

The result connects two questions: how accurately fractions can approximate $π$ , and how an infinite sum involving sines behaves. The manuscript thus announces both the resolution of the exponent conjecture and convergence of the Flint–Hills series.

2 · Going further

The problem concerns the irrationality exponent of $π$ , stated to equal 2. The supplied quantitative formulation is: for every $ε > 0$ , there is a threshold beyond which every positive integer denominator $q$ and every integer numerator $p$ satisfy $∣ π - p / q ∣ \geq q^{- 2 - ε}$ . The order of these conditions matters: first fix $ε$ , then take $q$ sufficiently large.

The manuscript abstract presents this equality as resolving a conjecture. It then announces, as a consequence, convergence of the Flint–Hills series $\sum_{n \geq 1} 1/ (n^{3} sin^{2} n)$ , with angles measured in radians. The connection between the two results is asserted, but the steps of the argument are not explained in the supplied text.

The main result has a Lean formalization. Its detailed scope is not provided here, particularly regarding convergence of the series. Only one manuscript is supplied: no companion manuscripts or argument methods are described.

Useful notions

Fractions and denominators
Absolute value as a measure of distance
Powers and small positive parameters
Convergence of a series
Sine and radian measure

What was known before / what is new

Before

The manuscript describes the assertion that the irrationality exponent of $π$ equals 2 as a conjecture. It calls the Flint–Hills series classical. The supplied text gives neither earlier bounds nor a detailed history.

New

The main result is that the irrationality exponent of $π$ equals 2, with the bound on rational approximations stated above. The manuscript derives convergence of the Flint–Hills series as a consequence. The main result has a Lean formalization.

Limits and assumptions

The bound requires $ε > 0$ and sufficiently large denominators; it does not assert the same inequality with $ε = 0$ or for every small denominator. Angles in the series are in radians. A Lean formalization of the main result is indicated, but its exact scope is not specified, so coverage of every announced consequence should not be assumed. The supplied text does not describe the method or identify any remaining open questions.

Mini glossary

Rational approximation: An approximation of a number by a fraction $p / q$ , with accuracy measured here by the distance $∣ π - p / q ∣$ .
Irrationality exponent: A number characterizing how accurately fractions with large denominators can approximate a number, stated here to equal 2 for $π$ .
Convergence of a series: The property that the sums obtained by successively adding terms approach a finite value.
Flint–Hills series: The infinite sum $\sum_{n \geq 1} 1/ (n^{3} sin^{2} n)$ , with angles in radians.

Keywords : Rational approximation · Irrationality exponent · Flint–Hills series

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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 $∣ π - p / q ∣ \geq q^{- 2 - ε}$ . This also proves convergence of the Flint–Hills series $\sum_{n \geq 1} 1/ (n^{3} sin^{2} n)$ , with angles in radians.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/017.md

Manuscripts in this family

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The irrationality exponent of pi is 2

24/09/2026

To be verified

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

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