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Erdős ProblemsNumber theory

Erdős Problem 269: Rational

Let PP be a finite set of primes with ∣P∣≥2|P| \ge 2 and let {a1<a2<… }\{a_1 < a_2 < \dots\} be the set of positive integers whose prime factors are all in PP. Is the sum ∑n=1∞1[a1,…,an]\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]} rational?

Number theoryAMS 11

Mathematical statement

Let PP be a finite set of primes with ∣P∣≥2|P| \ge 2 and let {a1<a2<… }\{a_1 < a_2 < \dots\} be the set of positive integers whose prime factors are all in PP. Is the sum ∑n=1∞1[a1,…,an]\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]} rational?

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_269.variants.rational

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem erdos_269.variants.rational : answer(sorry) ↔    ∀ᵉ (P : Finset ℕ) (h : ∀ p ∈ P, p.Prime) (h_card : P.card ≥ 2),    ∃ (q : ℚ), q = (series (P : Set 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 sourceformalizationerdosproblems.com/269background