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Erdős ProblemsNumber theory

Erdős Problem 321: Is Big O

Let R(N)R(N) be the size of the largest A⊆{1,...,N}A\subseteq\{1, ..., N\} such that all sums ∑n∈S1n\sum_{n\in S} \frac{1}{n} are distinct for S⊆AS\subseteq A. Find the simplest g(N)g(N) such that R(N)=O(g(N))R(N) = O(g(N)).

Number theoryAMS 11

Mathematical statement

Let R(N)R(N) be the size of the largest A⊆{1,...,N}A\subseteq\{1, ..., N\} such that all sums ∑n∈S1n\sum_{n\in S} \frac{1}{n} are distinct for S⊆AS\subseteq A. Find the simplest g(N)g(N) such that R(N)=O(g(N))R(N) = O(g(N)).

Statement source: Erdős Problems 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

erdos_321.variants.isBigO

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem erdos_321.variants.isBigO :    (fun N ↦ (R N : ℝ)) =O[atTop] (answer(sorry) : ℕ → ℝ) := 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 sourceformalizationerdosproblems.com/321background