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Claim. For every $k>2$ there is $n_0(k)$ such that, for a finite set $A\subset\mathbb C$ with $|A|\ge n_0(k)$, the multiset $A_k$ of sums of $k$ distinct elements of $A$, together with $|A|$, determines $A$. This is the theorem of Section 4 of B. Gordon, A. S. Fraenkel and E. G. Straus, On the determination of sets by the sets of sums of a certain order: with $F_s(n)$ the largest number of $n$-element multisets in a torsion-free abelian group that share one multiset $P_s$ of $s$-fold sums of distinct-index elements, "if $s>2$ then there is only a finite number of $n$ for which $F_s(n)>1$", a conjecture of Selfridge and Straus. Sets of complex numbers are multisets in the torsion-free group $(\mathbb C,+)$, and two distinct sets with the same $A_k$ would be two members of one class, so $F_k(n)=1$ gives the uniqueness; Section 2 shows that $F_s(n)$ is unchanged when the elements are restricted to positive integers. The proof rewrites the Selfridge--Straus condition for $F_s(n)>1$ as the Diophantine equation

$$\sum_{i\ge1}(-1)^{i-1}\binom{n}{s-i}\,i^{k-1}=0$$

(Section 3), locates its $s-1$ real roots in $n$ for large $k$ near $(s-j)(1+1/j)^{k-1}$, $1\le j\le s-1$, and, since every integer solution has $n\mid(s-1)!\,s^{k-1}$, applies Ridout's theorem on approximation by integers with prime factors in a fixed finite set to exclude infinitely many solutions when $s>2$. The method gives no explicit $n_0(k)$; the paper remarks that a Davenport--Roth argument would bound the number of exceptional $n$, far from best possible. The statement and the proof are recorded on the source card (claims checked; the proof checked for structure only, not verified).

The statement it settles. The theorem is the corrected Statement of Problem 494, which asks whether, for $k>2$, $A_k$ and $|A|$ determine $A$, provided $|A|$ is sufficiently large in terms of $k$; the problem page's Notes give the evidence for that form and the small sizes at which the site's wording fails. The formal-conjectures file states the theorem as the variant ∀ k > 2, ∀ᶠ card in atTop, Erdos494Unique k card (category research solved, sorry body, no formal-proof pointer) at its commit of 2026-09-18. The finite exceptional set is reported exactly for $k=3$, $|A|\in\{3,6,27,486\}$: Guy's 2004 collection, section C5, reports the triples problem settled by Boman and Linusson with exactly those exceptions, but prints the examples for $27$ and $486$ as multisets with repeated elements, the one for $27$ misprinted as given; for sets of distinct numbers the two large exceptions rest on the credit to Fomin and Izhboldin (1994) in the formal-conjectures statement file. For $k=4$, $|A|\in\{4,8\}$ are exceptional, $|A|=12$ is left in doubt by Selfridge and Straus, and Guy reports the four-sums problem settled by Ewell (Canad. J. Math. 1968) without listing its exceptions; Selfridge and Straus's page records those cases.

Depends on. Selfridge and Straus's claim page: Theorem 4 there gives the necessary condition for two distinct sets to share $A_k$, namely $F_s(n)>1$ forces $f(n,k)=0$ for some $k\le n$, which Section 3 of this paper rewrites as the Diophantine equation above and Section 4 bounds.

Acceptance. Refereed publication: Pacific Journal of Mathematics 12 (1962), no. 1, 187--196, received 29 March 1961, issued March 1962 (the Crossref record's date, the date of this page; the article's cover prints January 1962). Reviewed: the site's curator (T. F. Bloom) labels the problem PROVED and credits Gordon, Fraenkel and Straus with the uniqueness for all $k$ once $|A|$ is sufficiently large (erdosproblems.com/494, page last edited 14 October 2025), which is the corrected Statement, and Guy's 2004 collection reports the problem in its section C5. The site's label carries no Lean suffix and no formalization of this theorem is known (the 2026 Lean package on the Selfridge--Straus cases notes that Ridout's theorem is not in Mathlib), so no formalized evidence is listed. The development Erdos494.lean in Boris Alexeev's repository, whose header names Gordon, Fraenkel and Straus as informal authors, does not prove this page's theorem; it has its own claim page.