Erdős Problem #124 — open problem in number theory

Erdős Problem #124

Let k ≠ 0 and 3≤ d_1 < d_2 < ⋯ < d_r be integers of gcd equal to 1 such that Σ_1 ≤ i ≤ rfrac 1d_i - 1 ≥ 1. Can all sufficiently large integers be written as a sum of the shape Σ_i c_ia_i where c_i ∈ 0, 1 and a_i is divisible by d_i ^ k and has only the digits 0, 1 when written in base d_i?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-124,
  title        = {Erdős Problem #124},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-124}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $k \neq = 0$ and $3 \leq d_{1} < d_{2} < \dots < d_{r}$ be integers of gcd equal to $1$ such that $1 \leq i \leq r \sum \frac{1}{d _{i} - 1} \geq 1.$ Can all sufficiently large integers be written as a sum of the shape $\sum_{i} c_{i} a_{i}$ where $c_{i} \in {0, 1}$ and $a_{i}$ is divisible by $d_{i}^{k}$ and has only the digits $0, 1$ when written in base $d_{i}$ ?

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

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«124». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem 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

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/124. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/124
[BEGL96] Burr, S. A. and Erdős, P. and Graham, R. L. and Li, W. Wen-Ching, Complete sequences of sets of integer powers. Acta Arith. (1996), 133-138.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.