Bleicher 1976 denominators egyptian fractions

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conjecture_3: States the conjecture, now Problem 305's question, that the least possible largest denominator D(N) is at most a constant times N (ln N) to the power one plus epsilon, for every positive epsilon.

conjecture_4: States the 1976 conjecture that for an infinite sequence with ratios of consecutive terms bounded below by a constant above one, the rationals representable as finite sums of distinct reciprocals of its terms never fill an interval; the conjecture Problem 355 asks about.

theorem_1: For a prime P, the least possible largest denominator of a distinct unit-fraction expansion of some a/P is at least P times the least integer not below log₂ P.

theorem_2: Every fraction a/N has a distinct unit-fraction expansion whose largest denominator is at most a constant times N times the cube of ln N.

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M. N. Bleicher, P. Erdős: Denominators of Egyptian fractions, J. Number Theory 8 (1976), 157--168, doi:10.1016/0022-314X(76)90098-6 (MR 53 #7925; Zentralblatt 328.10010).

For a/b written as a sum of distinct unit fractions 1/n_1 + ... + 1/n_k with n_1 < ... < n_k, let D(a,b) be the smallest possible value of n_k and D(b) the maximum of D(a,b) over 0 < a < b. Theorem 2 (p. 162) gives the upper bound D(N) <= K N (ln N)^3 for every N >= 2 and an absolute constant K, obtained from an algorithm designed to minimize the largest denominator rather than the number of terms (in contrast to the Fibonacci-Sylvester, Erdos 1950, Golomb, and Bleicher Farey and continued-fraction algorithms, whose bounds the introduction tabulates). Theorem 1 (p. 158) gives a lower bound for primes: D(P) >= P ceil(log_2 P), not strict as printed, proved by tracking which denominators in a minimal expansion are divisible by P. The authors report that both theory and computation point to D(N)/N peaking at prime N (p. 158). These are the bounds cited for problem 305 on the smallest possible largest denominator in an Egyptian fraction representation, and Conjecture 3 (p. 167), D(N) <= K(epsilon) N (ln N)^{1+epsilon} for every epsilon > 0, is that problem's question.

The exponent of Theorem 2 is 3 as printed here. The site's commentary, the 1980 monograph of Erdős and Graham (p. 38) and Liu and Sawhney (2024) all attribute the bound D(b) << b (log b)^2 to this paper; the theorem with exponent 2 is Theorem 1 (p. 602) of part II, Denominators of Egyptian fractions II, Illinois J. Math. 20 (1976), 598--613, which proves D(N) <= lambda^3(N) N (ln N)^2 with lambda(N) -> 1 and whose introduction in turn recalls this paper's bound with exponent 4. Part II has its own card in this subject folder, Denominators of Egyptian fractions II.

The copy read for this card is a scan of the twelve printed pages whose OCR layer garbles the formulas; the statements below were read on the page images. Read status: claims checked. Theorems 1 and 2 and Conjectures 3 and 4 were read clause by clause on the page images of pp. 158, 162 and 167, the algorithm table of p. 157, the closing remarks and the bibliography (p. 168) were read, and no proof was checked. Result pages: theorem_1, theorem_2, conjecture_3, conjecture_4. The scan prints "Copyright © 1976 by Academic Press, Inc. All rights of reproduction in any form reserved." on its first page (printed p. 157), every other right reserved.

Source: https://users.renyi.hu/~p_erdos/1976-09.pdf.

Bears on. #305; #355: Conjecture 4 (p. 167), that no sequence with n_{i+1}/n_i > c > 1 has finite distinct-reciprocal sums representing every rational in an interval, is the statement whose negation the problem asks about; the problem page records its status.

Results to transcribe.

• Theorem 1 (p. 158): For P prime, D(P) >= P ceil(log_2 P); the inequality is not strict.
• Theorem 2 (p. 162): D(N) <= K N (ln N)^3 for every N >= 2 and an absolute constant K, via an algorithm minimizing the largest denominator.
• Conjecture 3 (p. 167): For every epsilon > 0 there is K(epsilon) with D(N) <= K(epsilon) N (ln N)^{1+epsilon}; the question of problem 305.
• Conjecture 4 (p. 167): for an infinite sequence n_1 < n_2 < ... with n_{i+1}/n_i > c > 1, the rationals a/b representable as 1/n_{i_1} + ... + 1/n_{i_t} for some t cannot fill an interval (alpha, beta); "We conjecture not", best possible by Graham's 1964 paper if true; the statement problem 355 tests.
• Algorithm survey: Tabulates prior expansion algorithms and their bounds on k and n_k, including Erdős 1950 with k <= 8 ln b/ln ln b and n_k <= 4b^2 ln b/ln ln b for b large.
• Conjectural extremal case: the authors' theoretical and numerical grounds for expecting D(N)/N to peak at prime N (p. 158).

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