Irrationality of Catalan’s constant

1 · The essentials

Catalan’s constant is a number defined by a sum that continues indefinitely. Start with 1, subtract $1/9$ , add $1/25$ , and continue with the squares of odd numbers in the denominators, alternating the signs. The manuscript announces that this number is irrational. This means it cannot equal any fraction of two integers. The main result has a formalization in Lean, a tool for checking mathematics. However, the supplied excerpt contains only the statement of the result: it does not let us explain the argument or specify the scope of that formalization.

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

As an imperfect image, imagine a length you want to measure exactly using a unit divided into a whole number of equal pieces. Calling the length irrational means that no whole number of those pieces will give the exact length, however you choose the division.

Why it matters

The result concerns the exact nature of Catalan’s constant, rather than an approximation to its value. It rules out every representation as a fraction of two integers. The Lean formalization provides formal verification of the main result, but the supplied material does not detail its contents.

2 · Going further

The problem concerns the constant defined by the series $G = \sum_{j \geq 0} (- 1)^{j} / (2 j + 1)^{2}$ . The index $j$ runs over the integers starting at zero; the factor $(- 1)^{j}$ alternates the signs, while the denominator is the square of the odd integer $2 j + 1$ . The irrationality statement says that there are no integers $p$ and $q$ , with $q \neq = 0$ , such that $G = p / q$ .

The repository summary and the abstract of “Catalan's constant is irrational” both announce this same result. The abstract supplies no construction, estimate, or main idea of the argument, so offering one here would be unjustified. The main result has a Lean formalization, but the supplied material does not describe its scope. Only one manuscript is presented; no relationship to companion manuscripts or historical context is given.

Useful notions

Fractions of integers
Infinite sums
Squares of odd numbers
Alternating signs

What was known before / what is new

Before

The source gives the definition of Catalan’s constant as an alternating infinite sum. It describes no earlier results about its irrationality or the history of the problem, so the prior state of knowledge cannot be reconstructed from this excerpt.

New

The announced contribution is the irrationality of Catalan’s constant: no fraction of two integers equals it exactly. The main result has a Lean formalization. The abstract describes no new method or additional result.

Limits and assumptions

The statement concerns precisely the constant G defined by the given series, with no additional hypotheses announced. A Lean formalization of the main result is reported, but its exact scope is not supplied: this should not be taken to mean that every possible claim in the manuscript is covered. The excerpt does not provide the argument and mentions no generalization, application, or remaining open question.

Mini glossary

Catalan’s constant: The number denoted by G and defined by the infinite sum $\sum_{j \geq 0} (- 1)^{j} / (2 j + 1)^{2}$ .
Irrational number: A number that equals no fraction $p / q$ where p and q are integers and q is nonzero.
Alternating series: An infinite sum whose terms successively switch signs.
Lean formalization: An expression of a mathematical statement and its proof in a language that allows Lean to check them.

Keywords : Catalan’s constant · Irrationality · Lean formalization

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 Catalan's constant $G = \sum_{j \geq 0} (- 1)^{j} / (2 j + 1)^{2}$ is irrational.

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/005.md

Manuscripts in this family

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Catalan's constant is irrational

24/09/2026

To be verified

We prove that Catalan's constant $G = \sum_{j \geq 0} (- 1)^{j} / (2 j + 1)^{2}$ is irrational.

PDF GitHub folder