Result 005, Number theory
Proves that Catalan's constant is irrational.
The bigger picture
Catalan's constant comes from a simple infinite sum, yet its exact arithmetic nature is the question here. The manuscript claims it cannot equal any fraction of integers, regardless of how large the denominator is.
Catalan's constant is formed by adding the reciprocals of successive odd squares, with alternating signs: one, minus one ninth, plus one twenty-fifth, and so on. The manuscript reports a proof that this specific number is irrational, meaning it is not a ratio of two integers. The claim concerns this constant alone, not a general class of infinite sums, and the abstract states no additional assumptions.
If established, the result would rule out an eventually repeating decimal expansion for Catalan's constant. It would also mean that no nonzero integer multiple of the constant can be an integer, excluding an exact arithmetic relationship rather than merely testing many candidate fractions. Irrationality alone does not, however, give quantitative bounds on how closely fractions can approximate it.
The immediate value is foundational: the claimed result would classify Catalan's constant more precisely within the real numbers and provide an arithmetic fact for subsequent arguments involving it. The supplied abstract does not describe a computational method or practical application. In particular, the claim should not be read as a faster way to calculate the constant's digits.
This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.
We prove that Catalan's constant is irrational.
@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}
}
OpenAI's note on what the formalization covers, from lean/docs/005.md.
The following describes the scope of the Lean formalization related to the following accompanying paper(s):
The formalization proves that Catalan's constant is irrational. This is the mathematical assertion in the paper's title.
| Result | Comparator statement |
|---|---|
| Irrationality of Catalan's constant | Catalan.lean |