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Claim. The answer to Problem 763 is no: for no $A\subseteq\mathbb N$ and no constant $c>0$ is $\sum_{n\le N}1_A*1_A(n)=cN+O(1)$. The claimed result is Theorem 1 of P. Erdős and W. H. J. Fuchs, On a problem of additive number theory: for an increasing sequence $0<a_1<a_2<\cdots$ of positive integers let $r(n)$ count the pairs $(i,j)$ with $a_i+a_j\le n$; then for $c>0$ the relation

$$r(n)=cn+o\bigl(n^{1/4}(\log n)^{-1/2}\bigr)$$

cannot hold. That is the journal's form of the theorem, as the zbMATH review of the paper (Zbl 0070.04104, by S. Selberg) and the site's commentary give it and as the formal-conjectures variant erdos_763.variants.erdos_fuchs states it; the 1954 Cornell technical-report printing that preceded the journal paper prints the weaker error term $o(n^{1/4}(\log n)^{-1/2-\varepsilon})$, $\varepsilon>0$. The paper introduces the theorem as the proof of the Erdős–Turán conjecture that $r(n)-cn=O(1)$ cannot hold, which is the site's question with $r(N)=\sum_{n\le N}1_A*1_A(n)$ counting ordered pairs; the paper notes that the result holds equally for the counts over $i<j$, $i\le j$ or all pairs, and for sequences of nonnegative reals. The method is a generating-function argument: with $g(z)=\sum_kz^{a_k}$, the assumed asymptotic is integrated against $g(z)^2$ on a circle of radius close to $1$ and contradicted by a lemma bounding such contour integrals. This page rests on the statement and the opening remarks of the 1954 printing, as the source card records; the proof pages of that scan are legible, but no proof check is recorded. Montgomery and Vaughan, after unpublished work of Jurkat, extended the impossibility to an error term $o(N^{1/4})$ (their claim page).

Depends on. Nothing in this wiki.

Acceptance. Refereed publication: J. London Math. Soc. 31 (1956), no. 1, 67--73, doi:10.1112/jlms/s1-31.1.67; the Crossref record dates the issue to January 1956, filled to the first of the month for this page's name. The 1954 printing is the Cornell University and Air Force Office of Scientific Research technical report of August 1954 that preceded the journal paper; the journal's pagination comes from the Crossref record and the zbMATH review, not from that printing. Reviewed: the site's curator, Thomas Bloom, labels the problem DISPROVED and credits the answer, in its strong form, to Erdős and Fuchs in the problem page's commentary (the proof-claim tab is empty and the thread has no posts). A Lean 4 development, src/latest/ErdosProblems/Erdos763.lean of Boris Alexeev's lean-proofs repository (1,495 lines at the pinned commit of 2026-09-15, first added 2026-08-17), declares itself a formalization of the solution: its header names Erdős, Fuchs, Montgomery and Vaughan as informal authors and Codex and GPT-5.6 Sol as formal authors, and its not_erdos_763 proves that for no A : Set ℕ and c > 0 is the summatory ordered representation count through N equal to c * N up to O(1), the bounded-error case only, by a Parseval argument on a circle; it closes with #print axioms not_erdos_763 without the printed output. The formal-conjectures statement for the problem (commit of 2026-09-20) is tagged solved and names line 1464 of the file, the theorem, as its formal proof; it also states the Erdős–Fuchs and Montgomery–Vaughan error terms as variants without proofs. The corpus has not built the development, so the page lists no formalized evidence.