OP-6961Status unassessedNumber theory

Waldschmidt's Conjecture on the Distance from $e^n$ to the Nearest Integer

TheoremDB P9326 · Candidate awaiting source review

Inferred from the community field; finer classification has not been reviewed.

Source candidate · not reviewedAs of 2026-09-08Source & evidence ↓

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Problem statement

For a real number $x$ , let $\lVert x \rVert$ denote the distance from $x$ to the nearest integer, defined as $\lVert x \rVert = \min_{m \in \mathbb{Z}} |x - m|$ . Prove that there exists a positive real number $c$ such that for every integer $n \geq 2$ , the inequality $\lVert e^n \rVert > n^{-c}$ holds.

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Formal statement

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theorem waldschmidt :
    ∃ c : ℝ, 0 < c ∧ ∀ n : ℕ, 2 ≤ n → (n : ℝ) ^ (-c) < distToNearestInt (Real.exp n)

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Progress & source assessment

Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.

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References

[1]TheoremDB P9326 link
[2]Source formal statement (not locally verified) link

Recorded progress

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Machine attempts

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Discussion

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