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Claim. Let $\{x\}=\{x_1,\ldots,x_n\}$ be complex numbers and $\{\sigma\}$ the multiset of sums of $s$ distinct elements. Theorem 4 of J. L. Selfridge and E. G. Straus, On the determination of numbers by their sums of a fixed order: if $n$ satisfies none of the equations $f(n,k)=0$, $k=1,\ldots,n$, where $f(n,k)$ is the coefficient of the power sum $S_k=\sum_ix_i^k$ in the expansion of $(s-1)!\sum_i\sigma_i^k$, an explicit polynomial in $n$ and $2^k,\ldots,s^k$, then the first $n$ power sums of $\{\sigma\}$ determine those of $\{x\}$ recursively, and hence $\{x\}$. Its corollary shows that every solution has $n\mid(s-1)!\,s^{n-1}$, so $\{x\}$ is determined whenever $n$ has a prime factor larger than $s$. For $s=2$, Theorems 1 and 2 give the exact answer: $\{x\}$ is determined when $n$ is not a power of two, and for $n=2^k$ distinct sets with the same pair sums exist, built inductively by translating half of a smaller pair. For $s=3$ (Example 1 and Theorem 5), $f(n,k)=n^2-(2^k+1)n+2\cdot3^{k-1}$ vanishes only for $k\in\{1,2,3,5,9\}$, at $n\in\{1,2,3,6\}$, where uniqueness fails in general, and at $n=27$ and $n=486$, which the paper leaves in doubt, so uniqueness holds for $n>6$ other than $27$ and $486$. For $s=4$ (Example 2 and Theorem 6) the roots are $n\in\{1,2,3,4,8,12\}$, uniqueness failing for the first five and $n=12$ left in doubt, so it holds for $n>12$. Theorem 3 shows that for $n>s$ a nontrivial transformation preserving $\{\sigma\}$ exists only for $n=2s$, so $|A|=2k$ is exceptional for every $k$, and Theorem 7 constructs arbitrarily large $s$ with a root $n>2s$. The statements of Theorems 1 to 6 are recorded on the source card (claims checked; proofs not verified). The paper poses the question in the form Problem 494 asks it, following a problem of L. Moser.

Covers. The corrected Statement of Problem 494 for $k=3$ and $k=4$: uniqueness holds for $k=3$ at every size $|A|>6$ other than $27$ and $486$, and for $k=4$ at every size $|A|>12$, so the exceptional sizes are finite in number for these $k$. For every $k>2$ it also covers the sizes $|A|$ with a prime factor greater than $k$, answered yes; and the companion case $k=2$, outside the problem's range, answered exactly: yes for $|A|$ not a power of two and no for $|A|$ a power of two. It does not show, for $k\ge5$, that the remaining sizes are finite in number, which the theorem of Gordon, Fraenkel and Straus on its page proves without listing them. The two doubtful sizes $27$ and $486$ for $k=3$ are reported as genuine exceptions, with a qualification on the sources: Guy's 2004 collection, section C5, reports the triples problem settled by Boman and Linusson with exceptions exactly $3$, $6$, $27$ and $486$, but prints the examples for $27$ and $486$ as multisets with repeated elements, and the one for $27$ is misprinted as given (the multiset and its negative have different element sums, so their triple sums cannot agree); for sets of distinct numbers, which the problem concerns, the two exceptions rest on the credit to Fomin and Izhboldin (1994) in the formal-conjectures statement file linked above. The site's commentary states the $k=3$ case for all $|A|>6$, without the two exceptions. Theorem 3, the failure at $|A|=2k$, answers only the site's wording, which the corrected Statement replaces.

Depends on. Nothing in this wiki.

Acceptance. Refereed publication: Pacific Journal of Mathematics 8 (1958), no. 4, 847--856, received 16 May 1958, issued December 1958 (the Crossref record's date, the date of this page; the article's cover prints June 1958). No reviewed evidence is listed: the site's curator (T. F. Bloom) mentions Selfridge and Straus's cases in the problem page's commentary, without the $27$ and $486$ caveat (erdosproblems.com/494, page last edited 14 October 2025), but the site's PROVED label settles the problem by crediting Gordon, Fraenkel and Straus, whose 1962 paper builds on this one and shares an author with it; the formal-conjectures statement file states the cases as solved variants with the caveat, with sorry bodies. Two Lean developments formalize parts of the result and are linked above; neither was built by this corpus, so no formalized evidence is listed. Collin Yuanjie Ren's package JSP-000399 (2026-09-16), whose README says it was prepared with Claude Code (Anthropic) assistance and credits the mathematics to Selfridge and Straus, proves the corollary to Theorem 4 (uniqueness when a prime greater than $k$ divides $|A|$) and Theorem 1 ($k=2$, $|A|$ not a power of two) in the statement file's own terms, and says that the Gordon--Fraenkel--Straus theorem is not formalized because Ridout's theorem is not in Mathlib; the community database (teorth/erdosproblems) notes the package under the problem. The fork of formal-conjectures that the statement file names as the formal proof of its variant k_eq_2_card_pow_two proves Theorem 2 ($k=2$, $|A|$ a power of two) and the counterexample variants at $|A|=k$ and $|A|=2k$ (Theorem 3), the last also recorded on the problem page. The development Erdos494.lean in Boris Alexeev's repository (added 2026-08-16; Codex and GPT-5.6 Sol as formal authors; file at a pinned commit) also proves the $|A|=2k$ case of Theorem 3, as the formal-conjectures variant card_eq_2k; it concerns the site's wording only and is credited on the problem page.