Problem 350

If $A\subset\mathbb{N}$ is a finite set of integers which is dissociated (that is, all of the subset sums are distinct) then

\sum_{n\in A}\frac{1}{n}<2.

Source: erdosproblems.com/350

Number theory
Additive combinatorics

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

Proved. The statement is Theorem 1 of Benkoski and Erdős (Math. Comp. 28 (1974), 617--623, refereed), whose proof the paper credits to C. Ryavec and reproduces in full; the refinement $\sum1/a_i\le2-2^{1-n}$ with the equality case is printed after the proof (p. 619), and Erdős's 1975 and 1977 surveys restate the theorem as Ryavec's proof of his February 1973 conjecture. The stronger bound $\sum_{n\in A}n^{-s}<1/(1-2^{-s})$ of Hanson, Steele and Stenger (Proc. Amer. Math. Soc. 66 (1977), 179--180, refereed) is not held, and its statement is quoted from the site and from the 1980 monograph. The site's label is PROVED (LEAN); its Lean mark dates from Alexeev's Lean file of 25 November 2025, and the formal-conjectures statement carries formal_proof attributes pointing to two external Lean files, Alexeev's, whose header credits the original proof to Ryavec, and a fork's proof added in April 2026 under the catalog's docstring; both are described below, and neither is Lean this corpus built. The claim pages are Benkoski and Erdős (Ryavec's proof; accepted on the refereed publication and the site's credit; the two Lean developments are its formalization links) and Hanson, Steele and Stenger (the stronger bound; accepted on the refereed publication, the note not held).

BeEr74: Benkoski, S. J. and Erdős, P., On weird and pseudoperfect numbers. Math. Comp. 28 (1974), no. 126, 617--623, doi:10.1090/S0025-5718-1974-0347726-9 (Crossref record). Theorem 1, p. 617, proof pp. 617--619, the refinement p. 619; Theorem 2, p. 619. Library home: Benkoski and Erdős 1974; result pages Theorem 1 and Theorem 2.
HSS77: Hanson, F., Steele, J. M. and Stenger, F., Distinct sums over subsets. Proc. Amer. Math. Soc. 66 (1977), no. 1, 179--180, doi:10.1090/S0002-9939-1977-0447167-4 (Crossref record; its deposited abstract reads "A finite set of integers with distinct subset sums has a precisely bounded Dirichlet series"). Not held. The statement is quoted from [ErGr80], p. 60, and the site.
Er75b: Erdős, P., Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (1974), Astérisque 24--25 (1975), 295--310; Chapter II, printed p. 302. Library home: Erdős 1975.
Er77c: Erdős, P., Problems and results on combinatorial number theory. III. Number theory day (Rockefeller Univ., 1976), Lecture Notes in Math. 626, Springer (1977), 43--72; Section 4, printed p. 52. Library home: Erdos 1977.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980); printed p. 60. Library home: Erdős and Graham 1980.

Problem 350

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