Konyagin 2014 double exponential lower bound number representations

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theorem_1: States the bound |X_n| at least exp(exp(((ln 2)(ln 3)/3 + o(1)) n / ln n)) for the number of representations of 1 by n distinct unit fractions, with the monotonicity inequality (1), Corollary 1, and two defects in the printed proof: a false identity in the proof of Lemma 2 and the failure of Lemma 1's second inequality for some even m.

theorem_2: States that, for large x, at most (x/ln x) exp((ln 2)^{-1}(ln ln ln x)^2) integers up to x occur as tau(2^n - 1) with n at most x, improving the bound x/(ln x)^{0.258} that the paper attributes to Luca and Shparlinski.

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Konyagin, S. V., Double exponential lower bound for the number of representations of unity by Egyptian fractions. Math. Notes 95 (2014), no. 1--2, 277--281.

The copy read for this card is the five-page Russian original, a short communication in Mat. Zametki 95 (2014), no. 2, 312--316, doi:10.4213/mzm10417. The English translation the site cites, Math. Notes 95 (2014), no. 1--2, 277--281, doi:10.1134/S0001434614010295, published online 28 February 2014, was not obtained and its labels were not compared (MathNet lists the translation's pages as 280--284). Read status: Theorem 1, inequality (1), display (2), Corollary 1 and Theorem 2 were read clause by clause on the PDF pages (claims checked); the proofs on pp. 313--315 were read for structure. Two steps of the printed proof of Theorem 1 fail: the displayed identity before its equation (4) is false, and the second inequality of Lemma 1 is false for m = 4; the Theorem 1 page records both, with the correction reported on the site. Nothing has been independently reviewed. The copy read prints "© С. В. Конягин, 2014" (S. V. Konyagin) at the foot of its first page (printed p. 312, read on the page image) and no license wording on its five pages; the Math-Net.Ru Terms of Use (https://www.mathnet.ru/php/agreement.phtml?option_lang=eng, read 2026-10-02) state "All materials published on this website including full-text articles, abstracts and author indexes are fully copyrighted by Steklov Mathematical Institute, Russian Academy of Sciences, and/or by other copyright holder" and "Reproduction or republication of the materials contained on Math-Net.Ru in any form requires written permission of the copyright holder", allow printing for noncommercial teaching or research only and name no open license, every other right reserved.

Let X_n be the set of representations 1 = 1/x_1 + ... + 1/x_n with 1 ≤ x_1 < ... < x_n; the map replacing x_n by x_n+1 and x_n(x_n+1) shows |X_n| ≤ |X_{n+1}|, and the previously known bounds were exp(c_1 n^3 / ln n) ≤ |X_n| ≤ exp((c_2+o(1)) 2^n) with c_2 < 0.12. Theorem 1 replaces the singly exponential lower bound by a doubly exponential one: |X_n| ≥ exp(exp(((ln 2)(ln 3)/3 + o(1)) n / ln n)) as n → ∞, so the count of Egyptian-fraction representations of unity grows at least doubly exponentially in n/ln n. The construction splits the last term of the paper's representation (4) of 1 by 3k+2 unit fractions in one way for each divisor of $(2^m+1)^2$ below $2^m+1$, exploiting the many divisors of $(2^m+1)^2$ supplied by primitive prime factors (Lemma 1, p. 313) to gain a second exponential. As a second application of Lemma 1, Theorem 2 (p. 315) bounds by (x/ln x) exp((ln 2)^{-1}(ln ln ln x)^2), for large x, the number of integers up to x of the form tau(2^n - 1) with n at most x. The paper is a short communication in Russian. For Erdős problem 148, which asks for good estimates of the number of ways to write 1 as a sum of n distinct unit fractions, Theorem 1 is a doubly exponential lower bound, against upper bounds of the form exp(O(2^n)).

Source: https://www.mathnet.ru/eng/mzm10417.

Bears on. #148: Theorem 1 (p. 312) states a lower bound for |X_n|, which is the problem's count F(n); it does not settle the problem, which asks for good estimates of F(n), and its printed proof has the two defects the Theorem 1 page records. Inequality (1) gives F(n) <= F(n+1).

Results.

• Theorem 1 (p. 312): As n → ∞, |X_n| ≥ exp(exp(((ln 2)(ln 3)/3 + o(1)) n / ln n)), where X_n is the set of representations of 1 as a sum of n distinct unit fractions; the page also records Corollary 1 (p. 314), the same bound for every positive rational, and two defects of the printed proof: a false identity in the proof of Lemma 2 and the failure of Lemma 1's second inequality at m = 4.
• Inequality (1) (p. 312, recorded on the Theorem 1 page): |X_n| ≤ |X_{n+1}| for all n, via the injection {x_1,...,x_n} ↦ {x_1,...,x_{n-1}, x_n+1, x_n(x_n+1)}.
• Theorem 2 (p. 315): for large x, at most (x/ln x) exp((ln 2)^{-1}(ln ln ln x)^2) integers in [1, x] are values tau(2^n - 1) with n ≤ x.

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