The board / number theory / #221
Erdős Problem #221
PROVED LEANdatabase updated 2026-01-31
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
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Is there a set \(A\subset\mathbb{N}\) such that, for all large \(N\), \[\lvert A\cap\{1,\ldots,N\}\rvert \ll N/\log N\] and such that every large integer can be written as \(2^k+a\) for some \(k\geq 0\) and \(a\in A\)? Lorentz [Lo54] proved there is such a set with, for all large \(N\), \[\lvert A\cap\{1,\ldots,N\}\rvert \ll \frac{\log\log N}{\log N}N\] The answer is yes, proved by Ruzsa [Ru72].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/221.Prizenone
Topicsnumber theory, additive basis
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/221
Attempts
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