Montgomery 1979 solution problem e2689 egyptian fractions

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solution_p224: Hahn's problem E2689 as reprinted with its solution, and Montgomery's two sets of positive integers, each a union of separated blocks of at least two consecutive integers, whose reciprocals sum to 2; the second set is the five-interval example Problem 289 cites.

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L.-S. Hahn (proposer) and Peter L. Montgomery (solver), E2689, under the heading "Egyptian Fractions" in Elementary Problems and Solutions, Amer. Math. Monthly 86 (1979), no. 3, p. 224, DOI 10.2307/2321534; the problem was proposed in Amer. Math. Monthly 85 (1978), p. 47 (the head of the solution prints "E 2689 [1978, 47]"). The proposer is placed at the University of New Mexico and the solver at the System Development Corporation, Huntsville, Alabama (p. 224). Cited as [Mon79] on the problem page, where [Hah78] is the proposal; the 1980 Erdős--Graham monograph's [Hah (78)] and [Mon (79)] are the same two items (its p. 34, filed as erdos_1980_old_new_problems_results_combinatorial_number_theory). The item has no title of its own beyond the problem number and the section heading; the problem page cites it as "Solution to Problem E2689".

The copy read for this card is JSTOR's scan of the printed page, the version of record: 2 pages, PDF p. 1 a JSTOR cover sheet (citation, stable URL, access date) and printed p. 224 = PDF p. 2. The printed page carries the end of the solution to a preceding lattice-point problem with its editor's comment, then E2689 (problem, solution and a one-line note on Hickerson), then the start of problem E2691; each of the two pages ends in JSTOR's download footer, which carries the download date and time and the downloading machine's address (text layer, PDF pp. 1--2), and the copy's metadata recorded only its 2026 assembly. The scan has a text layer that reads the prose and the two sets cleanly and garbles the displayed condition (2) (the sum sign and the fraction come out as scattered characters). Provenance: the copy read was obtained on 2026-09-22 from JSTOR, the DOI https://doi.org/10.2307/2321534 resolving to the stable URL https://www.jstor.org/stable/2321534 and its PDF; 219,685 bytes. No copyright line is printed on the page; the JSTOR cover sheet and page footers read "All use subject to https://about.jstor.org/terms", whose Terms and Conditions of Use (https://about.jstor.org/terms/, read 2026-10-02) state that the intellectual property in the content is proprietary to its contributors, allow downloading only "in reasonable amounts for non-commercial, scholarly purposes" and prohibit providing access to non-authorized users, and the JSTOR item page (https://www.jstor.org/stable/2321534) could not be read on 2026-10-02, every other right reserved.

Read status: claims checked for the whole E2689 item, the reprinted problem with its conditions (1) and (2), the solution's two sets and its closing sentence, and the note on Hickerson, each read clause by clause on the page image of PDF p. 2 (printed p. 224) on 2026-09-22; the cover sheet (PDF p. 1) was read on the page image for the citation. The item prints no argument beyond the two sets, so nothing was read for structure only. The two reciprocal sums were recomputed here in exact rational arithmetic and both equal 2; nothing here is independently reviewed.

Contents

• The problem (p. 224, page image). Hahn asks whether there is a nonempty finite set $S$ of positive integers such that (1) every $n\in S$ has $n-1\in S$ or $n+1\in S$, and (2) the sum of $1/n$ over $n\in S$ is an integer. Condition (1) says that $S$ splits into maximal runs of consecutive integers, each of length at least two; two maximal runs are never adjacent, since adjacent runs would form one run. So $S$ is a union of separated blocks of at least two consecutive integers, with no prescribed number of blocks, and (2) asks for any integer, not a specified one.
• The solution (p. 224, page image). Montgomery answers yes and gives two sets: $S_1=\{1,2,7,8,13,14,39,40,76,77,285,286\}$, six blocks of length two, and $S_2=\{2,3,4,5,6,7,9,10,17,18,34,35,84,85\}$, the blocks $[2,7]$, $[9,10]$, $[17,18]$, $[34,35]$ and $[84,85]$. Quoted (p. 224): "In each case, the Egyptian fractions sum to 2." No derivation is printed. Both sums were recomputed here and equal $2$ exactly.
• The note (p. 224, page image), quoted: "The second example was also found by Dean Hickerson." The first set is Montgomery's alone as printed. The page prints no list of other solvers and no editor's comment for E2689.

Compiled scope

The item is compiled at statement depth for what Problem 289 consumes: the reprinted problem and the two sets, read on the page image and paged on solution_p224. The proposal itself, Amer. Math. Monthly 85 (1978), p. 47, was not consulted; its text is known here only as reprinted at the head of the solution. Nothing here is independently reviewed.

Bears on. #289: the solution (printed p. 224, PDF p. 2) is the primary source of the example the site and the 1980 monograph attribute to it. $S_2$ is the five-interval representation $2=\sum_{i=1}^5\sum_{n\in I_i}1/n$ with $I_1=[2,7]$, $I_2=[9,10]$, $I_3=[17,18]$, $I_4=[34,35]$, $I_5=[84,85]$ that the site's commentary quotes, and the monograph's fourteen denominators are this set. The page settles the attribution the problem page had from secondary sources: the solution is Montgomery's, and the note credits the second set to Hickerson as well, so "Hickerson and Montgomery" (the site) fits $S_2$ and "Montgomery" (the monograph) fits the solution. It also confirms that Hahn's problem asked for any integer, not for $1$, and that its condition (1) is the separated-block condition of the site's formulation with the number of blocks free. The first set $S_1$, not mentioned on the site or in the monograph, is a six-interval representation of $2$ with every block of length exactly two. The item proves nothing about representing $1$ and nothing about all large $k$, so the problem's status is unchanged.

Results.

• Solution (p. 224): the two sets $S_1$ and $S_2$, each satisfying Hahn's conditions (1) and (2) with reciprocal sum $2$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.