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Claim. Christian Elsholtz and Stefan Planitzer, Sums of four and more unit fractions and approximate parametrizations, Bull. Lond. Math. Soc. 53 (2021), no. 3, 695--709, arXiv:2012.05984v1 (10 December 2020), which dates this page. Their Corollary 3(2) (arXiv v1, p. 5): let $u_0=1$, $u_{n+1}=u_n(u_n+1)$ and $c_0=\lim_{n\to\infty}u_n^{2^{-n}}=1.5979102\ldots$ (their Remark 3); then for every $\varepsilon>0$ and $k\ge k(\varepsilon)$,

$$f_k(1,1)\ <\ c_0^{(2/5+\varepsilon)2^{k-1}},$$

where $f_k(1,1)$ counts the nondecreasing $k$-tuples of positive integers with reciprocal sum $1$. The distinct increasing solutions are among them, so $F(k)\le f_k(1,1)$, and since $c_0=E^2$ for the Vardi constant $E=1.264084\ldots$, the bound reads $F(k)<E^{(2/5+\varepsilon)2^k}$. The corollary comes from their Theorem 2, $f_k(m,n)\ll_\varepsilon(kn)^\varepsilon(k^{4/3}n^2/m)^{(8/5)2^{k-5}}$ for $k\ge5$, and the paper refers its proof to the authors' earlier paper and to Browning and Elsholtz. The statement and its locators are on the [[../library/unit_fractions/elsholtz_2021_sums_four_more_unit_fractions_approximate/corollary_3|result page]] of the [[../library/unit_fractions/elsholtz_2021_sums_four_more_unit_fractions_approximate/_index|source card]].

Normalization. As printed, the bound is weaker than the earlier Browning--Elsholtz bound $E^{(5/24+\varepsilon)2^k}$, which the authors' 2020 paper (card; arXiv:1805.02945v1, p. 1) states with $c_0=1.264\ldots$ and $u_1=1$. The successive improvements change only the exponent constant of the lifted bound, $5/3$ (Browning and Elsholtz), $28/17$ (2020) and $8/5$ (this paper), and with $c_0=E$ these give $5/24$, $7/34$ and $1/5$ as the coefficient of $2^k$; the 2020 paper's own Corollary 3(2) also prints $u_0=1$, which would likewise make its bound weaker than the one it improves. Read with $u_1=1$, as the site's commentary reads it, Corollary 3(2) gives $F(k)<E^{(1/5+\varepsilon)2^k}$. That reading is a deduction of this page; no source it cites corrects the printed normalization.

Covers. An upper bound for $F(k)$ of the form $c^{2^k}$: $F(k)<E^{(2/5+\varepsilon)2^k}$ for $k\ge k(\varepsilon)$ as printed, and $E^{(1/5+\varepsilon)2^k}$ under the reading above. It gives no lower bound, no asymptotic formula and no estimate up to constant factors.

Depends on. Nothing in this wiki.

Acceptance. Refereed: the paper is published in the Bulletin of the London Mathematical Society. The site's commentary credits the paper with the upper bound, but the site labels the problem OPEN, so that commentary is not listed as review.