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Claim. The answer to Problem 764 is no: for no $A\subseteq\mathbb N$ and no constant $c>0$ is $\sum_{n\le N}1_A*1_A*1_A(n)=cN+O(1)$. The claimed result is the theorem of R. C. Vaughan, On the addition of sequences of integers, in the form the site's commentary records: for $c>0$ the three-fold representation count through $N$ cannot equal $cN$ with an error $o(N^{1/4}(\log N)^{-1/2})$, the three-summand case of a general theorem on $h$-fold convolutions that allows other main terms; it extends the Erdős–Fuchs theorem for two summands, the subject of Problem 763. The paper is not held; the statement follows the site's commentary and the formal-conjectures docstring.

Depends on. Nothing in this wiki.

Acceptance. Refereed publication: J. Number Theory 4 (1972), no. 1, 1--16, doi:10.1016/0022-314X(72)90008-X; the Crossref record dates the issue to February 1972, filled to the first of the month for this page's name. Reviewed: the site's curator, Thomas Bloom, labels the problem disproved and credits the answer, in its strong form, to Vaughan in the problem page's commentary (empty proof-claim tab and no thread posts). A Lean 4 development, src/latest/ErdosProblems/Erdos764.lean of Boris Alexeev's lean-proofs repository (3,873 lines at the pinned commit of 2026-09-15, first added 2026-08-17), declares itself a formalization of the negative answer: its header names Vaughan as informal author and Codex and GPT-5.6 Sol as formal authors, cites this paper, and describes its proof as the bounded-error specialization of Vaughan's argument, by the differentiated generating-function identity, Fourier orthogonality on a circle and a geometric kernel; its not_erdos_764 proves that for no A : Set ℕ and c > 0 is the summatory ordered three-fold convolution through N equal to c * N up to O(1), the bounded-error case only, and it closes with #print axioms Erdos764.not_erdos_764 without the printed output. The formal-conjectures statement for the problem (commit of 2026-09-20) is tagged solved and names line 3760 of the file, the theorem, as its formal proof, and states Vaughan's error term as a variant without proof. The corpus has not built the development, so the page lists no formalized evidence.