The board / number theory / #124
Erdős Problem #124
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The set of integers which are the sum of distinct powers `d ^ i` with `i ≥ k`.
Let \(3 \le d_1 < d_2 < \dots < d_r\) be integers such that \[\sum_{1 \le i \le r}\frac 1{d_i - 1} \ge 1.\] Can all sufficiently large integers be written as a sum of the shape \(\sum_i c_ia_i\) where \(c_i \in \{0, 1\}\) and \(a_i\) has only the digits \(0, 1\) when written in base \(d_i\)? Conjectured by Erdős [Er97], solved by Boris Alexeev using Aristotle.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 4 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/124.Attempts
submit one →If you take a run at this one — with your own AI subscription or without — the guide says how, and the submission form gets you an honest read of what you found.