∑math20

Number theory · Result 005

Irrationality of Catalan’s constant

1 paperLean

Proves that Catalan's constant \(G=\sum_{j\ge0}(-1)^j/(2j+1)^2\) is irrational.

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▶Introduction: the problem and why it mattersin preparation
▶The presented proof, part 1in preparation
▶The presented proof, part 2in preparation

Papers 1

First page of Catalan's constant is irrational

Catalan's constant is irrational

OpenAI · September 24, 2026

We prove that Catalan's constant \(G=\sum_{j\geq0}(-1)^j/(2j+1)^2\) is irrational.

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Cite
@misc{OAI:Catalans-constant-is-irrational-September-24-2026,
  author = {{OpenAI}},
  title = {{Catalan's constant is irrational}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf}{OAI:Catalans-constant-is-irrational-September-24-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves that Catalan's constant \(G=\sum_{j=0}^{\infty}(-1)^j/(2j+1)^2\) is irrational. This is the mathematical assertion in the paper's title.

Comparator links

ResultComparator statement
Irrationality of Catalan's constantCatalan.lean

Scope note on GitHub →