The board / number theory / #14
Erdős Problem #14
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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The number of integers in \(\{1,\ldots,N\}\) which are not representable in exactly one way as the sum of two elements from \(A\) (either because they are not representable at all, or because they are representable in more than one way).
Let \(A ⊆ \mathbb{N}\). Let \(B ⊆ \mathbb{N}\) be the set of integers which are representable in exactly one way as the sum of two elements from \(A\). Is it true that for all \(\epsilon > 0\) and large \(N\), \(|\{1,\ldots,N\} \setminus B| \gg_\epsilon N^{1/2 - \epsilon}\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/14.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA143824
Original pageerdosproblems.com/14
Attempts
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exact finite exception minima E(N) are exhaustively verified for every 1<=N<=80, with explicit witnesses and a 43,083,585-node from-scratch checker; the asymptotic problem remains open.
PARTIAL