Problem 494

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claims/: The 3 claim pages of Problem 494, one per claimant's result; the problem's standing derives from them.

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Statement. If $A\subset \mathbb{C}$ is a finite set and $k\geq 1$ then let

$$A_k = \{ z_1+\cdots+z_k : z_i\in A\textrm{ distinct}\}.$$

For $k>2$ does the multiset $A_k$ (together with the size of $A$) uniquely determine the set $A$?

Statement (corrected). If $A\subset \mathbb{C}$ is a finite set and $k\geq 1$ then let

$$A_k = \{ z_1+\cdots+z_k : z_i\in A\textrm{ distinct}\}.$$

For $k>2$ does the multiset $A_k$ (together with the size of $A$) uniquely determine the set $A$, provided $|A|$ is sufficiently large in terms of $k$?

Notes. The site's wording sets no range for $|A|$, and as printed the answer is no for every $k>2$: for $|A|<k$ the multiset $A_k$ is empty and for $|A|=k$ it is the single sum of $A$, so distinct sets of those sizes share it (Kruyt's observation, recorded in the site's commentary); for $|A|=2k$ a set with sum $0$ and its negative share $A_k$, since each $k$-sum of the negative is minus a $k$-sum of $A$, which is the complementary $k$-sum (Tao's observation, in the commentary and in Tao's forum comment of 30 August 2025, which says the case "needs to be ruled out"; Theorem 3 of [SeSt58], p. 850, shows $n=2s$ is the only size above $s$ at which a nontrivial transformation preserves the $s$-fold sums); for $k=3$ the sizes $27$ and $486$ are reported as further exceptions (see Formulation). Bloom reads the problem with a size condition. The commentary records the two failures as remarks on the wording, says "Presumably some condition like '$|A|$ sufficiently large' is intended", and labels the problem PROVED on the theorem of Gordon, Fraenkel and Straus [GFS62] that "for all $k$, the multiset $A_k$ uniquely determines $A$ provided $|A|$ is sufficiently large", added on 14 October 2025 after msellke's forum comment of 13 October 2025 supplied the reference. The corrected Statement adopts that reading in Bloom's words, adding "provided $|A|$ is sufficiently large in terms of $k$" and changing nothing else. It is the conjecture Gordon, Fraenkel and Straus attribute to Selfridge and Straus [GFS62, §1, p. 187], that for $s>2$ one has $F_s(n)=1$ "for all but a finite number of $n$", which for fixed $k$ says the same thing. Under the site's wording the answer is no, by the small sizes above; under Bloom's reading it is yes, by the theorem of Section 4 of [GFS62], with the explicit ranges $k=3$, $|A|>6$ other than $27$ and $486$, and $k=4$, $|A|>12$, from [SeSt58]. The missing range is older than the site: Erdős's report of the problem ([Er61], item I.33, p. 238) states the Selfridge--Straus conjecture for $k>2$ with no size condition (and with products in place of sums; see Formulation), while Selfridge and Straus ask "To what extent is $\{x\}$ determined by $\{\sigma\}$" ([SeSt58], §1, p. 847) and call the exceptional pairs $(s,n)$ "in a certain sense quite rare" (p. 854). Results on the site's wording are credited here and do not bear on the standing: Kruyt's failure at $|A|=k$, Tao's at $|A|=2k$, Theorem 3 of [SeSt58], and the Lean 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), whose only theorem, card_eq_2k, proves the failure at $|A|=2k$ for every $k>2$ and whose own section heading reads "The literal problem has a negative answer". The theorem answers the site's wording, not the corrected Statement, which asks only about sufficiently large $|A|$; it presents itself as a solution of Problem 494 and so has a claim page, which is rejected and does not count toward the standing. The page's standing judges the corrected Statement.

Formulation. In [Er61] (item I.33, p. 238) Erdős states the problem of Selfridge and Straus with products of $k$ of the numbers in place of sums; the site calls this a misstatement and follows the sums of [SeSt58], which Erdős cites. The product form is false: for $k=3$ the sets of sixth roots of unity with exponents $\{0,1,2,4\}$ and $\{0,2,3,4\}$ have the same products of three distinct elements (exponent sums $\{0,1,3,5\}$ modulo $6$ in both cases), an example the site credits to Steinerberger. The exceptional sizes for $k=3$ are reported as exactly $3$, $6$, $27$ and $486$ by [Gu04], whose examples for $27$ and $486$ are printed as multisets with repeated elements (the one for $27$ misprinted as given), so for sets of distinct numbers the two large exceptions rest on the credit to Fomin and Izhboldin (1994) in the formal-conjectures statement file; the exceptional sizes are finite in number for every $k>2$ by [GFS62].

Status. The site labels the problem PROVED and credits Gordon, Fraenkel and Straus with uniqueness for every $k$ once $|A|$ is sufficiently large; the corrected Statement is Bloom's reading, so the label describes it. The theorem of Section 4 of [GFS62] (Pacific J. Math. 12 (1962), 187--196, refereed) settles it: for every $k>2$, all but finitely many sizes $|A|$ admit no two distinct sets with the same multiset $A_k$. Selfridge and Straus [SeSt58] settle $k=3$ for $|A|>6$ other than $27$ and $486$, $k=4$ for $|A|>12$, and every $k$ when $|A|$ has a prime factor exceeding $k$. The claim pages are Gordon, Fraenkel and Straus (an accepted full claim, on the refereed publication and the site's credit) and Selfridge and Straus (an accepted partial claim).

Source. erdosproblems.com/494, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #494, https://www.erdosproblems.com/494.

References.

• [Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221--254; item I.33, printed p. 238.
• [GFS62] Gordon, B. and Fraenkel, A. S. and Straus, E. G., On the determination of sets by the sets of sums of a certain order. Pacific J. Math. 12 (1962), no. 1, 187--196.
• [Gu04] Guy, Richard K., Unsolved problems in number theory, third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp.; C5 "Sums determining members of a set", printed pp. 167--168: Moser's question of how far the pair sums determine a set, settled by Selfridge, Straus and others when the cardinality is not a power of two; the sums-of-triples problem settled by Boman and Linusson with exceptions exactly at sizes 3, 6, 27 and 486; and the sums of four distinct elements settled by Ewell. Library home: guy_2004_unsolved_problems_number_theory.
• [SeSt58] Selfridge, J. L. and Straus, E., On the determination of numbers by their sums of a fixed order. Pacific J. Math. 8 (1958), no. 4, 847--856.

Formalization. Statement in formal-conjectures at its commit of 2026-09-18: the file states the Selfridge--Straus cases, the counterexamples at $|A|=k$ and $|A|=2k$ and the large-size theorem as variants with sorry bodies, and has no unqualified main statement; the large-size variant is the corrected Statement. The counterexample variants and the $k=2$ power-of-two variant point to formal proofs in a fork of the repository (commit of 2026-03-12), linked on Selfridge and Straus's claim page together with Collin Yuanjie Ren's Lean package of 2026-09-16 for the two positive Selfridge--Straus criteria, which the community database (teorth/erdosproblems) notes under the problem. The development src/latest/ErdosProblems/Erdos494.lean of Boris Alexeev's lean-proofs repository (first added 2026-08-16; formal authors Codex and GPT-5.6 Sol; informal authors named as Gordon, Fraenkel and Straus) calls itself a formalization of a solution, but its only theorem, card_eq_2k, is the formal-conjectures variant that uniqueness fails at $|A|=2k$ for every $k>2$. It has its own claim page, rejected because it answers the site's wording, not the corrected Statement (its only theorem is the variant card_eq_2k, the failure at $|A|=2k$), and the Notes credit it. No formalization of the Gordon--Fraenkel--Straus theorem is known, and this corpus has built none of these developments, so no formalized evidence is listed.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• gordon_1962_determination_sets_sets_sums_certain_order
• gordon_1962_determination_sets_sets_sums_certain_order / section_5
• gordon_1962_determination_sets_sets_sums_certain_order / theorem_p190
• selfridge_1958_determination_numbers_sums_fixed_order
• selfridge_1958_determination_numbers_sums_fixed_order / corollary_p853
• selfridge_1958_determination_numbers_sums_fixed_order / theorem_1
• selfridge_1958_determination_numbers_sums_fixed_order / theorem_2
• selfridge_1958_determination_numbers_sums_fixed_order / theorem_3
• selfridge_1958_determination_numbers_sums_fixed_order / theorem_4
• selfridge_1958_determination_numbers_sums_fixed_order / theorem_5
• selfridge_1958_determination_numbers_sums_fixed_order / theorem_6
• selfridge_1958_determination_numbers_sums_fixed_order / theorem_7
• erdos_1961_unsolved_problems
• guy_2004_unsolved_problems_number_theory