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Erdős ProblemsNumber theory

Erdős Problem 124: Ne Zero

Let k≠0k \ne 0 and 3≤d1<d2<⋯<dr3\leq d_1 < d_2 < \cdots < d_r be integers of gcd equal to 11 such that ∑1≤i≤r1di−1≥1.\sum_{1 \le i \le r}\frac 1{d_i - 1} \ge 1. Can all sufficiently large integers be written as a sum of the shape ∑iciai\sum_i c_ia_i where ci∈{0,1}c_i \in \{0, 1\} and $a...

Number theoryAMS 11

Mathematical statement

Let k≠0k \ne 0 and 3≤d1<d2<⋯<dr3\leq d_1 < d_2 < \cdots < d_r be integers of gcd equal to 11 such that ∑1≤i≤r1di−1≥1.\sum_{1 \le i \le r}\frac 1{d_i - 1} \ge 1. Can all sufficiently large integers be written as a sum of the shape ∑iciai\sum_i c_ia_i where ci∈{0,1}c_i \in \{0, 1\} and aia_i is divisible by dikd_i ^ k and has only the digits 0,10, 1 when written in base did_i?

Conjectured by Burr, Erdős, Graham, and Li [BEGL96]

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

erdos124.ne_zero

Canonical source
Complete statement target, proof intentionally absentLean 4
lemma erdos124.ne_zero : answer(sorry) ↔    ∀ k ≠ 0, ∀ D : Finset ℕ, (∀ d ∈ D, 3 ≤ d) → 1 ≤ ∑ d ∈ D, (d - 1 : ℚ)⁻¹ → D.gcd id = 1 →      ∀ᶠ n in atTop, n ∈ ∑ d ∈ D, sumsOfDistinctPowers d k := 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/124background