Ruderman 1971 e2232 representation 1 egyptian fractions problem

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conjecture_p303: Erdős and Straus state that they consider it certain that the least number of distinct unit fractions summing to one with denominators at least n exceeds (e−1)n by an amount tending to infinity, and that they have not proved it.

inequality_1: Bounds the least number of distinct unit fractions summing to one with every denominator at least n, between (e−1)n minus a constant and (e−1)n plus a constant times n over log n.

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H. D. Ruderman, P. Erdos, E. Straus, E2232 (Representation of 1 by Egyptian Fractions), Problem and solution. The American Mathematical Monthly 78 (1971), no. 3, 302-303. doi:10.2307/2317539.

Ruderman's problem E2232 asks for an upper bound on U_n, the least number of distinct unit fractions summing to 1 whose largest term is at most 1/n (with U_1 = 1, U_2 = 3, U_3 = 5). The published solution by Erdos and Straus proves the two-sided estimate (1): there are constants c_1, c_2 with (e-1)n - c_2 < U_n < (e-1)n + c_1 n / log n. The lower bound follows immediately from comparing the harmonic sum with a logarithm; the upper bound chooses m with 1/n + ... + 1/m < 1 < 1/n + ... + 1/m + 1/(m+1), notes m = en + O(1), writes the deficit as u/v with v at most the lcm of integers up to m so v < e^(2m), and then applies Erdős's 1950 theorem (display (5), cited by title without a theorem number; it is Theorem 1 of that paper) that u/v is a sum of fewer than c log v / log log v distinct unit fractions; the print infers from (2), (3) and (5) that their denominators exceed m + 1. The solvers add (p. 303) that it seems certain to them that U_n - (e-1)n tends to infinity, but they have not proved it; the editorial note after the solution records a conjecture of several solvers that 2n is an upper bound. Problem 295's site cites this solution; the constant e - 1 and both halves of -c < U_n - (e-1)n << n / log n that the site records come from it.

Source: https://www.jstor.org/stable/2317539.

The copy read for this card is the JSTOR scan of the two printed pages with a cover sheet; its text layer garbles the displays, which were read on the page images. Read status: claims checked. The definition, the examples, inequality (1) and displays (2)-(5) were read clause by clause on the page images, and the short proof and the remark and editorial note of p. 303 were read in full; see inequality_1 and conjecture_p303. The published Erdős--Straus solution was therefore read in full. No copyright line is printed on the pages; 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/2317539) could not be read on 2026-10-02, every other right reserved.

Bears on. #295: inequality (1) is the pair of bounds -c < k(N) - (e-1)N << N / log N that the problem records, and does not decide whether the excess tends to infinity; the unproved remark of p. 303 states that divergence as the authors' belief.

Results to transcribe.

• Inequality (1) (p. 302): There are constants c_1, c_2 with (e-1)n - c_2 < U_n < (e-1)n + c_1 n / log n.
• Construction (2)-(4): Taking m maximal with 1/n + ... + 1/m < 1 gives m = en + O(1), and the leftover u/v < 1/(m+1) has v < m^(pi(m)) < e^(2m).
• Cited theorem of Erdős (5) (p. 303): For 1 <= u < v, u/v is a sum of fewer than c log v / log log v distinct unit fractions (Mat. Lapok 1 (1950), 192-210, cited without a theorem number; it is Theorem 1 of that paper); the print infers x_1 > m + 1 from (2), (3) and (5).
• Remark (p. 303): The authors state it seems certain that U_n - (e-1)n tends to infinity but give no proof; the editorial note records that several solvers conjecture 2n is an upper bound for U_n.

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