Source labels openChecked July 26, 2026

Erdős Problem 457: One Sub

Taking $n$ to be the product of primes between $\log n$ and $(2 + o(1)) \log n$ gives an example where $q(n, \log n) \ge (2 + o(1)) \log n.$ Can one prove that $q(n, \log n) < (1 - \epsilon) (\log n)^2$ for all large $n$ and some $\epsilon > 0$ ?

Number theory AMS 11

Mathematical statement

Taking $n$ to be the product of primes between $\log n$ and $(2 + o(1)) \log n$ gives an example where

q(n, \log n) \ge (2 + o(1)) \log n.

Can one prove that $q(n, \log n) < (1 - \epsilon) (\log n)^2$ for all large $n$ and some $\epsilon > 0$ ?

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_457.variants.one_sub

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_457.variants.one_sub : answer(sorry) ↔ ∃ ε > (0 : ℝ),    ∀ᶠ n in Filter.atTop, q n (Real.log n) < (1 - ε) * Real.log n ^ 2 := 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 erdosproblems.com/457background