theoremnovelinferred_dep

Irrationality exponent of pi is 2

Irrationality exponent of

\pi

is 2:

|\pi-p/q|\ge q^{-\nu}

for large

q

the headline result

Theorem·thm:main·§1 Introduction

Statement (paper)

The irrationality exponent of

\pi

2

. More precisely, for every real

\nu>2

, there is an integer

Q(\nu)

such that

\left|\pi-\frac pq\right|\ge q^{-\nu} \qquad(p\in\ZZ,\ q\in\ZZ,\ q\ge Q(\nu)).

In plain English

For every real

\nu>2

there is an integer

Q(\nu)

such that

|\pi-p/q|\ge q^{-\nu}

for all integers

p

and

q\ge Q(\nu)

, so the irrationality exponent of

\pi

equals

2

Why it's here

The upper bound is proved by contradiction. Assuming good approximations with arbitrarily large

q

, the parameter lemma, interpolation theorem and the arithmetic and translation lemmas let all error terms be made small, and the determinant comparison then fails. The lower bound

2

is the easy Dirichlet pigeonhole argument, which needs irrationality of

\pi

, itself a consequence of the contradiction argument. The result is the paper's main theorem and feeds the Flint–Hills series corollaries.

Classified novel:

The claimed upper bound of 2 for the irrationality exponent of π is the paper’s main new result; known irrationality-exponent bounds do not establish this sharp value. The matching lower bound follows from Dirichlet’s approximation theorem.

Inside the proof

claimChoose F_0 so nu/F_0 is smallclaimDimension choice makes errors small, savings largeclaimApproximants chosen with center-independent weight thresholdsclaimCombined error margins contradict determinant comparisonclaimPi is irrationalclaimDirichlet pigeonhole gives lower bound exponent 2

Selected node

Statement (paper)

In plain English

Why it's here

Inside the proof