Claims

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1958_12_01_selfridge_straus: Selfridge and Straus (Pacific J. Math. 1958) determine a set from its sums of s distinct elements unless its size is a root of an explicit equation: k = 3 for n > 6 other than 27 and 486, k = 4 for n > 12; an accepted partial claim.

1962_03_01_gordon_fraenkel_straus: Section 4 of Gordon, Fraenkel and Straus (Pacific J. Math. 1962) proves, for every k > 2, that all but finitely many sizes |A| are determined by the k-fold sums, which settles the corrected statement; accepted, refereed.

2026_08_16_alexeev: Answers the site's wording (any size of A, which fails at size 2k), not the corrected Statement (all sufficiently large sizes), so it does not count toward the problem's standing. A Lean theorem in Alexeev's repository, written by Codex and GPT-5.6 Sol, proves that for every k > 2 two sets of size 2k share their k-fold sums.

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