The board / number theory / #1136
Erdős Problem #1136
PROVED LEANdatabase updated 2026-04-19
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
A set `A` of natural numbers has the property in the question if `a + b ≠ 2 ^ k` for all `a, b ∈ A` (not necessarily distinct) and all `k ≥ 0`.
The set of all integers congruent to \(3\cdot 2^i\pmod{2^{i+2}}\) for some \(i\geq 0\).
Does there exist \(A\subset \mathbb{N}\) with lower density \(>1/3\) such that \(a+b\neq 2^k\) for any \(a,b\in A\) and \(k\geq 0\)? Müller [Mu11] settled this question in the affirmative: in fact one can take \(A\) to be the set of all integers congruent to \(3\cdot 2^i\pmod{2^{i+2}}\) for any \(i\geq 0\), which has density \(1/2\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1136.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/1136
Attempts
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