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Erdős ProblemsNumber theory

Erdős Problem 458

Let lcm⁡(1,…,n)\operatorname{lcm}(1, \dots, n) denote the least common multiple of {1,…,n}\{1, \dots, n\}. Let pkp_k be the kk-th prime. Is it true that for all k≥1k \geq 1, lcm⁡(1,…,pk+1−1)<pk⋅lcm⁡(1,…,pk)\operatorname{lcm}(1, \dots, p_{k+1}-1) < p_k \cdot \operatorname{lcm}(1, \dots, p_k)?

Number theoryAMS 11

Mathematical statement

Let lcm⁡(1,…,n)\operatorname{lcm}(1, \dots, n) denote the least common multiple of {1,…,n}\{1, \dots, n\}. Let pkp_k be the kk-th prime. Is it true that for all k≥1k \geq 1, lcm⁡(1,…,pk+1−1)<pk⋅lcm⁡(1,…,pk)\operatorname{lcm}(1, \dots, p_{k+1}-1) < p_k \cdot \operatorname{lcm}(1, \dots, p_k)?

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_458

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem erdos_458 :    answer(sorry) ↔ ∀ k : ℕ, lcm_upto ((k + 1).nth Prime - 1)     < k.nth Prime * lcm_upto (k.nth Prime) := 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/458background