Proofmap
Family 017Number theory23 pp · Sep 24, 2026

The irrationality exponent of pi is 2

What this paper proves

For every real ν>2\nu>2, there is an integer Q(ν)Q(\nu) such that ∣π−p/q∣≥q−ν|\pi-p/q|\ge q^{-\nu} for every integer pp and every integer q≥Q(ν)q\ge Q(\nu); the irrationality exponent of π\pi is 22. A weighted-degree inequality for curves bounds their total contact height at prescribed points, enabling the determinant comparison that rules out better approximations. The result matches the Dirichlet lower bound and feeds the paper’s Flint–Hills series corollaries.

Main theorem

theoremnovelIrrationality exponent of pi is 2thm:main
For every real ν>2\nu>2 there is an integer Q(ν)Q(\nu) such that ∣π−p/q∣≥q−ν|\pi-p/q|\ge q^{-\nu} for all integers pp and q≥Q(ν)q\ge Q(\nu), so the irrationality exponent of π\pi equals 22.
Theorem 1.1 · as stated in the paper
The irrationality exponent of π\pi is 22. More precisely, for every real ν>2\nu>2, there is an integer Q(ν)Q(\nu) such that ∣π−pq∣≥q−ν(p∈Z, q∈Z, q≥Q(ν)). \left|\pi-\frac pq\right|\ge q^{-\nu} \qquad(p\in\ZZ,\ q\in\ZZ,\ q\ge Q(\nu)).

Where the new idea lives · 2 novel steps (one is the main theorem)

Proposition 2.3Curve inequality for weighted degree
Every curve has weighted degree at least (1+σ)(1+\sigma) times its total contact height

Shape of the proof

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40
statements mapped
14
sections
2
novel steps
57
friction flags
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