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Erdős ProblemsField theory and polynomials

Erdős Problem 477

Is there a polynomial f:Z→Zf:\mathbb{Z}\to \mathbb{Z} of degree at least 22 and a set A⊂ZA\subset \mathbb{Z} such that for any z∈Zz\in \mathbb{Z} there is exactly one a∈Aa\in A and b∈{f(n):n∈Z}b\in \{ f(n) : n\in\mathbb{Z}\} such that z=a+bz=a+b?

Field theory and polynomialsAMS 12

Mathematical statement

Is there a polynomial f:Z→Zf:\mathbb{Z}\to \mathbb{Z} of degree at least 22 and a set A⊂ZA\subset \mathbb{Z} such that for any z∈Zz\in \mathbb{Z} there is exactly one a∈Aa\in A and b∈{f(n):n∈Z}b\in \{ f(n) : n\in\mathbb{Z}\} such that z=a+bz=a+b?

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_477

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem erdos_477 : answer(sorry) ↔    ∃ f : ℤ[X], 2 ≤ f.degree ∧ ∃ A : Set ℤ,      ∀ z, ∃! ab ∈ A ×ˢ (f.eval '' {n | 0 < n}), z = ab.1 + ab.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 sourceformalizationerdosproblems.com/477backgroundSek59background