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Claim. S. V. Konyagin, Double exponential lower bound for the number of representations of unity by Egyptian fractions, Mat. Zametki 95 (2014), no. 2, 312--316, English translation Math. Notes 95 (2014), 277--281. The page is dated January 2014, the month Crossref gives for Math. Notes 95, no. 1--2. Theorem 1 (printed p. 312 of the Russian original): as $n\to\infty$,

$$F(n)\ \ge\ \exp\Bigl(\exp\Bigl(\Bigl(\frac{(\ln2)(\ln3)}{3}+o(1)\Bigr)\frac{n}{\ln n}\Bigr)\Bigr),$$

where the paper's $|\mathbf X_n|$ is the problem's $F(n)$. The paper also proves $F(n)\le F(n+1)$ (inequality (1), p. 312, by an explicit injection) and the same lower bound for every positive rational (Corollary 1, p. 314). The proof builds, for $n=3k+3$ and a suitable $m<3^k$, about $\tau((2^m+1)^2)/2$ distinct representations from the divisors of $(2^m+1)^2$, and uses the Bang--Zsigmondy theorem and a divisor-function bound to make that count doubly exponential. The statement, the proof's structure and the defect below are on the [[../library/unit_fractions/konyagin_2014_double_exponential_lower_bound_number_representations/theorem_1|result page]].

Defect. The displayed identity on p. 314 immediately before the paper's equation (4) is false (it fails at $k=2$, $m=1$). The defect was reported in the site's discussion thread on 5 September 2025, and a comment of 29 December 2025 reports the author's agreement, a corrected identity and a missing odd-$m$ hypothesis in Lemma 1, with the main result standing. The corrected argument is unpublished and unverified, and as of 2026-09-17 no erratum appears in the MathNet, Crossref or publisher records of either version. Independently of that step, [[../library/unit_fractions/elsholtz_2016_egyptian_fractions_odd_denominators/corollary_1_2|Elsholtz's Corollary 1.2]] (Q. J. Math. 67 (2016)) gives the doubly exponential order $\exp(\exp(ck/\log k))$ with an unspecified $c>0$, through odd denominators and without Konyagin's identities; the constant $(\ln2)(\ln3)/3$ rests on Konyagin's printed proof alone.

Covers. A lower bound for $F(k)$ of the form $\exp(\exp(ck/\log k))$. Its order is superseded by [[problems/unit_fractions/E0148/claims/2026_09_25_openai|the release's Corollary 1.2]], which gives $\log\log F(k)\ge ck$ for large $k$. It gives no upper bound and no asymptotic formula.

Depends on. Nothing in this wiki.

Acceptance. Refereed: the paper is published in Matematicheskie Zametki and its translation in Mathematical Notes. The page keeps the result accepted on that record with the defect in its printed proof stated above. The site's commentary credits the paper with the lower bound, but the site labels the problem OPEN, so that commentary is not listed as review.