ZNumber.lean`

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Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 0 contain sorry, 0 declared as axioms

Mahler's Z-numbers

A Z-number is a positive real number ξ such that the fractional part of ξ (3/2)^n lies in [0, 1/2) for every positive integer n. Mahler asked whether any Z-number exists; this is still open.

References:

[Mah68] Mahler, Kurt. "An unsolved problem on the powers of 3/2." Journal of the Australian Mathematical Society 8.2 (1968): 313-321.
Wikipedia, Mahler's 3/2 problem

Main definitions

IsZNumber: a real number is a Z-number.

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

def IsZNumber A Z-number is a positive real number ξ such that the fractional part of ξ (3/2)^n lies in [0, 1/2) for every positive integer n line 41 source
Source documentation
A Z-number is a positive real number ξ such that the fractional part of ξ (3/2)^n lies in [0, 1/2) for every positive integer n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: And, LT.lt, OfNat.ofNat, Nat, LE.le, Int.fract, HMul.hMul, HPow.hPow, HDiv.hDiv
23 instances, coercions and auxiliary constants left out.
```
def IsZNumber (ξ : ℝ) : Prop :=
…
```