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Erdős ProblemsNumber theory

Erdős Problem 830: Ii

Erdos Problem 830, Part 2* We say that a,b∈Na,b\in \mathbb{N} are an amicable pair if σ(a)=σ(b)=a+b\sigma(a)=\sigma(b)=a+b. If A(x)A(x) counts the number of amicable 1≤a≤b≤x1\leq a\leq b\leq x then is it true that A(x)>x1−o(1)?A(x) > x^{1-o(1)}?

Number theoryAMS 11

Mathematical statement

Erdos Problem 830, Part 2* We say that a,b∈Na,b\in \mathbb{N} are an amicable pair if σ(a)=σ(b)=a+b\sigma(a)=\sigma(b)=a+b. If A(x)A(x) counts the number of amicable 1≤a≤b≤x1\leq a\leq b\leq x then is it true that A(x)>x1−o(1)?A(x) > x^{1-o(1)}?

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_830.parts.ii

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem erdos_830.parts.ii : answer(sorry) ↔ ∃ o : ℝ → ℝ, o =o[atTop] (1 : ℝ → ℝ) ∧ ∀ᶠ x in atTop,    x ^ (1 - o x) < A x := 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/830background