The board / number theory / #764
Erdős Problem #764
DISPROVEDdatabase updated 2025-08-31
The statement as posed is false.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The number of representations of `n` as an ordered sum of three elements of `A`, \(1_A\ast 1_A\ast 1_A(n)\).
Let \(A\subseteq \mathbb{N}\). Can there exist some constant \(c>0\) such that \[\sum_{n\leq N} 1_A\ast 1_A\ast 1_A(n) = cN+O(1)?\] The case of \(1_A\ast 1_A(n)\) is the subject of [763](https://www.erdosproblems.com/763). The answer is no, proved in a strong form by Vaughan [Va72], who showed that in fact \[\sum_{n\leq N} 1_A\ast 1_A\ast 1_A(n) = cN+o\left(\frac{N^{1/4}}{(\log N)^{1/2}}\right)\] is impossible. Vaughan proves a more general result that applies to any \(h\)-fold convolution, with different main terms permitted.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/764.Attempts
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