The board / number theory / #869
Erdős Problem #869
DISPROVEDdatabase updated 2025-08-31
The statement as posed is false.
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If \(A_1, A_2\) are disjoint additive bases of order \(2\) (i.e. \(A_i + A_i\) contains all large integers) then must \(A = A_1 \cup A_2\) contain a minimal additive basis of order \(2\) (one such that deleting any element creates infinitely many \(n \notin A + A\))? A question of Erdős and Nathanson [ErNa88, Er92c]. The answer is no: Larsen [La26] constructed disjoint bases \(A_1, A_2\) of order \(2\) such that \(A_1 \cup A_2\) contains no minimal basis of order \(2\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/869.Prizenone
Topicsnumber theory, additive basis
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/869
Attempts
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