Erdos 1950 az egyenlet egesz szamu megoldasairol diophantine

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conjecture_p195: Records the 1950 statement, attributed to Erdős and Straus, that every fraction 4/b with b > 4 is a sum of at most three distinct unit fractions, with Straus's verification for b < 5000.

conjectures_p194: Records Erdős's 1950 conjectures on gaps and ratios among the denominators of a representation of one by distinct unit fractions and his question on the number of such representations.

theorem_1: Every fraction a/b with 0 < a < b is a sum of fewer than c₁ log b over log log b distinct unit fractions, with c₁ = 8 for b above 4096.

theorem_2: For every b the average of N(a,b) over a is more than half of log log b minus one, and N(b−1,b) itself exceeds log log b minus one.

theorem_4: Proves that the Sylvester sequence 2, 3, 7, 43, ... gives the largest possible last denominator of an n-term representation of one and the largest proper fraction representable by at most n unit fractions.

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P. Erdős: Az ${1\over x_1} + {1\over x_2} +\cdots + {1\over x_n} = {a\over b}$ egyenlet egész számú megoldásairól (On a Diophantine equation; in Hungarian, with Russian and English summaries on pp. 209--210), Mat. Lapok 1 (1950), 192--210; MR 13,280b.

This Hungarian survey-with-proofs studies N(a,b), the least n for which a/b is a sum of n distinct unit fractions with 0 < x_1 < ... < x_n, and reviews the history from the Rhind papyrus through Nakayama's work. Theorem 1 (1. tetel, p. 195) proves that there is a constant c_1 with N(a,b) < c_1 log b / log log b for all 0 < a < b (with c_1 = 8 for b > 4096, p. 202), sharpening de Bruijn's bound N(a,b) < c log b / log log log b, quoted on p. 195 without a reference; on the same page Erdős writes that probably N(a,b) < c' log log b for a suitable constant c', the question of problem 304. Theorem 2 (2. tetel, p. 195) gives complementary lower bounds: the average of N(1,b), ..., N(b-2,b) exceeds (1/2)(log log b - 1), and N(b-1,b) > log log b - 1; Erdős reads this as saying that relatively many of N(1,b), ..., N(b-2,b) exceed c_2 log log b. Erdős also records (p. 194) unsolved problems and conjectures for the unit equation with a = b: that every solution with distinct increasing x_i has some gap x_{i+1} - x_i >= 3 (Kürschák's theorem gives >= 2), that possibly for every c every solution with enough terms has some gap > c, that x_n/x_1 >= 3 with equality only for 2, 3, 6, that probably x_n/x_1 tends to infinity with n (both ratio statements contradicted by Croot's short-intervals theorem), and he asks for the number of solutions; he introduces the Sylvester-type recursion alpha_1 = 2, alpha_{n+1} = alpha_n(alpha_n - 1) + 1 (equivalently alpha_{n+1} = alpha_1 ... alpha_n + 1) giving 2, 3, 7, 43, ... whose reciprocals with a final 1/(alpha_n - 1) sum to 1. The paper is the source cited for Erdos problems 206 and 304 on representing rationals as sums of distinct unit fractions and on the structure of such representations.

Source: https://users.renyi.hu/~p_erdos/1950-02.pdf.

The copy read for this card is a scan of the nineteen printed pages whose OCR layer garbles the formulas; everything cited here was read on the page images. Read status: claims checked. Theorems 1-5, the conjectures and questions of p. 194, the 4/n conjecture passage of p. 195 with its English summary on p. 210, Nakayama's theorems as quoted and displays (4)-(11) were read clause by clause; the proofs of Theorem 1 (pp. 198-203), Theorem 2 (pp. 208-209) and Theorem 5 (pp. 205-208) were read for structure and are summarized on the result pages, not verified. Result pages: theorem_1, theorem_2, theorem_4 (Theorems 3-5 and the recursion), conjectures_p194, conjecture_p195 (the 4/n conjecture, made with Straus, with Straus's verification for 4 < b < 5000). No copyright or license line is printed (pp. 192--193 and 209--210 read); the hosting archive's site footer speaks for the site, not the paper ("(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only.", https://users.renyi.hu/~p_erdos/, read 2026-10-02); Matematikai Lapok has no publisher page for the 1950 volume, so none was consulted, and no Crossref license is recorded; the term is unstated.

Bears on. #206, #304, #287 (the p. 194 gap conjecture), #148 (the p. 194 counting question), #293 (Theorem 3 bounds every denominator of an n-term representation of 1 by alpha_n - 1), #295 (Theorem 1 is the input of the Erdős--Straus solution of Monthly problem E2232), #242 (p. 195: the conjecture, made with Straus, that N(4,b) <= 3 for every b > 4, that is, 4/b is a sum of at most three distinct unit fractions, with Straus's verification for 4 < b < 5000; the site's source key [Er50c])

Results to transcribe.

• 1. tetel / Theorem 1 (p. 195, eq. 5): There is a constant c_1 such that N(a,b) < c_1 log b / log log b for every 0 < a < b.
• 2. tetel / Theorem 2 (p. 195, eqs. 6-7): (1/(b-2)) sum_{a=1}^{b-2} N(a,b) > (1/2)(log log b - 1), and N(b-1,b) > log log b - 1, for every positive integer b.
• Elementary bound (p. 193, eq. 4): N(a,b) <= a always (eq. 4); Nakayama's criterion for N(a,b) = 2 is quoted (p. 193), and N(3,b) = 3 holds exactly when every prime factor of b is of the form 6k+1 (p. 194; the Hungarian text prints N(a,b) in this statement, the English summary on p. 210 states it for N(3,b)).
• Conjectures (p. 194): every solution of 1 = sum 1/x_i with x_1 < ... < x_n has some gap x_{i+1} - x_i >= 3 (unproved by Erdős; (2,3,6) shows 3 is best possible for n = 3); possibly for every c there is n_0 such that every solution with n > n_0 terms has some gap > c; x_n/x_1 >= 3 with equality only for (2,3,6); probably x_n/x_1 tends to infinity with n, that is, for every q every solution with enough terms has x_n > q x_1. Erdős also asks for the number f_1(n) of solutions in positive integers and the number f_2(n) of solutions with x_1 < ... < x_n.
• Sylvester recursion (p. 196, eqs. 8-11): The sequence alpha_1 = 2, alpha_{n+1} = alpha_n(alpha_n-1)+1 = alpha_1...alpha_n + 1 (2, 3, 7, 43, ...) satisfies sum_{i<n} 1/alpha_i + 1/(alpha_n - 1) = 1; Theorems 3-5 (pp. 197, 203) prove that this is the extremal solution, that every denominator of an n-term representation of 1 is at most alpha_n - 1, and that 1 - 1/(alpha_{n+1} - 1) is the largest proper fraction with N(a,b) <= n.

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