Normality of mathematical constants

π is normal in base 10.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-normality-of-pi,
  title        = {Normality of mathematical constants},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/normality-of-pi}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

$π$ is normal in base 10.

Despite extensive empirical evidence—billions of digits have been computed for $π$ , $e$ , and $2$ , all showing near-uniform digit distribution—it is an open problem whether any of the classical constants $π$ , $e$ , $2$ , $ln 2$ , or $φ$ is normal in any base.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.NormalityOfPi.

theorem pi_normal_base_ten : IsNormalInBase 10 π

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.