Selfridge 1958 determination numbers sums fixed order

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corollary_p853: Selfridge and Straus's corollary to Theorem 4 that every root n of f(n,k) = 0 divides (s-1)! s^{n-1}, so the sums of s distinct elements always determine the n numbers when s is less than the greatest prime factor of n.

theorem_1: Selfridge and Straus's theorem that, for sums of two distinct elements and n not of the form 2^k, the first n elementary symmetric functions of the pair sums can be prescribed arbitrarily and determine the n numbers uniquely.

theorem_2: Selfridge and Straus's theorem that, for sums of two distinct elements and n = 2^k, the power sums of the pair sums up to order k + 1 satisfy an algebraic equation and the pair sums do not always determine the n numbers.

theorem_3: Selfridge and Straus's theorem that, for n > s, a nontrivial transformation preserving the multiset of sums of s distinct elements exists only when n = 2s, and is then linear, with every diagonal entry equal to -(s-1)/s and every off-diagonal entry 1/s up to permutations.

theorem_4: Selfridge and Straus's theorem that, for every s, if n satisfies none of the Diophantine equations f(n,k) = 0 for k = 1, ..., n, the first n elementary symmetric functions of the sums of s distinct elements can be prescribed arbitrarily and determine the n numbers uniquely, while at a root the first k of them satisfy an algebraic equation.

theorem_5: Selfridge and Straus's theorem that for sums of three distinct elements the equation f(n,k) = 0 has solutions only for k = 1, 2, 3, 5, 9, with Example 1 listing the roots n = 1, 2, 3, 6, 27, 486 and leaving 27 and 486 in doubt.

theorem_6: Selfridge and Straus's theorem that for sums of four distinct elements the equation f(n,k) = 0 has solutions only for n = 1, 2, 3, 4, 8, 12, with Example 2 saying the sums do not generally determine the set at the first five and leaving n = 12 in doubt.

theorem_7: Selfridge and Straus's theorem that there are arbitrarily large s for which f(n,k) = 0 for some n > 2s, obtained from Pell equations for the quadratic factors of f(n,4) and f(n,5).

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Selfridge, J. L. and Straus, E. G., On the determination of numbers by their sums of a fixed order. Pacific J. Math. 8 (1958), no. 4, 847-856, doi:10.2140/pjm.1958.8.847.

Selfridge and Straus ask to what extent a set {x_1, ..., x_n} of complex numbers is determined by the multiset of sums of s distinct elements, a question suggested by a problem of L. Moser. For s = 2 they prove (Theorem 1) that if n is not a power of 2 the first n elementary symmetric functions of the sum-set can be prescribed arbitrarily and determine {x} uniquely, while (Theorem 2) if n = 2^k the power sums must satisfy an algebraic relation and {x} is not always determined; the counterexamples are built inductively by the translate construction {x_i + a} union {y_j} versus {x_i} union {y_j + a}. For general s they show (Lemma 1, Theorem 3) that for n > s any nontrivial transformation preserving the sum-set is linear and forces n = 2s, with an explicit matrix, and (Theorem 4) that {x} is determined uniquely unless n is a root of one of an explicit family of Diophantine equations f(n,k) = 0, k = 1, ..., n, coming from the coefficient of S_k in the expansion of the sum-set power sums; at a root the first k elementary symmetric functions of the sums satisfy an algebraic equation, and the paper finds non-uniqueness at some roots and leaves others in doubt. A corollary (p. 853) shows that every root n divides (s-1)! s^{n-1}, so {x} is determined whenever s is less than the greatest prime factor of n. Theorems 5 and 6 solve these equations for s = 3 (solutions only for k = 1, 2, 3, 5, 9, at n = 1, 2, 3, 6, 27, 486, the last two left in doubt) and s = 4 (solutions only for n = 1, 2, 3, 4, 8, 12, the last left in doubt), and Theorem 7 constructs arbitrarily large s for which f(n,k) = 0 for some n > 2s. The paper bears on problem 494 by locating the possible non-uniqueness cases for reconstructing a set from its s-fold subset sums, and by listing the possible exceptional sizes for s = 3 and s = 4.

Source: https://msp.org/pjm/1958/8-4/p17.xhtml. No notice is printed on the PDF's cover page or in the volume's back matter; the publisher's article pages print "(c) Copyright", the article's year and "Pacific Journal of Mathematics. All rights reserved.", a line read on another article's page (https://msp.org/pjm/1962/12-1/p17.xhtml, read 2026-10-02) and not on this article's own page, every other right reserved.

Read status: claims checked for Theorems 1 to 7, Lemma 1, the corollary to Theorem 4, Examples 1 and 2 and the definition (15) of $f(n,k)$, read clause by clause on the page images of the print; the proofs of Theorems 1, 2 and 5 followed, the others read for structure. Nothing here is independently reviewed.

Bears on. #494: the paper asks the problem's question for sums of $s$ distinct elements (p. 847). By Theorem 4 (p. 852) the sums together with $n$ determine the set unless $n$ is a root of one of the equations $f(n,k)=0$, $k=1,\ldots,n$, and by its corollary (p. 853) whenever $n$ has a prime factor greater than $s$. For $s=3$, Theorem 5 and Example 1 (p. 853) leave only $n=1,2,3,6,27,486$, with $27$ and $486$ in doubt; for $s=4$, Theorem 6 and Example 2 (p. 854) leave only $n=1,2,3,4,8,12$, with $12$ in doubt. Theorem 3 (p. 850) names $n=2s$ as the only size above $s$ admitting a nontrivial transformation that preserves the sums, and Theorem 7 (p. 855) gives arbitrarily large $s$ with a root $n>2s$, sizes the method leaves undecided. The paper does not show, for general $s$, that the exceptional sizes are finite in number. Theorem 1 and Theorem 2 settle the case $s=2$, which the problem leaves out.

Results.

• Theorem 1 (p. 847): for $s=2$ and $n\neq2^k$, the first $n$ elementary symmetric functions of the pair sums can be prescribed arbitrarily and determine $\{x\}$ uniquely.
• Theorem 2 (p. 848): for $s=2$ and $n=2^k$, $\Sigma_1,\ldots,\Sigma_{k+1}$ satisfy an algebraic equation and the pair sums do not always determine $\{x\}$.
• Theorem 3 (p. 850), with Lemma 1 (p. 849): if $n>s$ and a nontrivial transformation preserves the $s$-fold sums, then $n=2s$ and the transformation is linear with an explicit matrix up to permutations.
• Theorem 4 (p. 852): if $n$ satisfies none of the equations $f(n,k)=0$, $k=1,\ldots,n$, the first $n$ elementary symmetric functions of the sums can be prescribed arbitrarily and determine $\{x\}$ uniquely; if $f(n,k)=0$, the first $k$ of them satisfy an algebraic equation.
• Corollary (p. 853): if $f(n,k)=0$ then $n$ divides $(s-1)!\,s^{n-1}$, so $\{x\}$ is determined when $s$ is less than the greatest prime factor of $n$.
• Theorem 5 (p. 853), with Example 1: for $s=3$, $f(n,k)=0$ has solutions only for $k=1,2,3,5,9$, at $n=1,2,3,6,27,486$.
• Theorem 6 (p. 854), with Example 2: for $s=4$, $f(n,k)=0$ has solutions only for $n=1,2,3,4,8,12$.
• Theorem 7 (p. 855): there are arbitrarily large $s$ with $f(n,k)=0$ for some $n>2s$.

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