Source labels openChecked July 26, 2026

Bugeaud Collection of Conjectures and Open Questions: Fractional Parts of Powers

Problem 10.3. To prove that there exists a positive real number~ $c$ such that $\lVert e^n \rVert > e^{−cn}$ , for every~ $n \ge 1$ . Posed by Mahler [Mah53].

Number theory AMS 11

Mathematical statement

Problem 10.3. To prove that there exists a positive real number~ $c$ such that $\lVert e^n \rVert > e^{−cn}$ , for every~ $n \ge 1$ . Posed by Mahler [Mah53].

Statement source: Books statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

problem_10_3

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem problem_10_3 :    ∃ c : ℝ, 0 < c ∧ ∀ n : ℕ, 1 ≤ n → Real.exp (-c * n) < distToNearestInt (Real.exp n) := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization Wal03background