Problem 148

Let $F(k)$ be the number of solutions to

1= \frac{1}{n_1}+\cdots+\frac{1}{n_k},

where $1\leq n_1<\cdots<n_k$ are distinct integers. Find good estimates for $F(k)$ .

Source: erdosproblems.com/148

Number theory
Unit fractions

Wiki page

Status

Open

No claim settles this problem.

Open on the site: the label is OPEN (page last edited 27 September 2025; no proof claim on its tab). The order of the double logarithm is fixed: $ck\le\log\log F(k)\le Ck$ for all large $k$ , by Corollary 1.2 of the OpenAI mathematics release's manuscript of 25 September 2026 (its claim page (OpenAI, 2026)); the lower half is new and replaces the published $\exp(\exp(ck/\log k))$ , the upper half was known from explicit bounds of the form $c^{2^k}$ , the best of them due to Browning and Elsholtz and to Elsholtz and Planitzer (see Upper bound below). No asymptotic formula, no estimate up to constant factors and no value of the constant in $\log\log F(k)$ is known.

ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), p. 32.
Ko14: Konyagin, S. V., Double exponential lower bound for the number of representations of unity by Egyptian fractions. Math. Notes 95 (2014), 277--281, doi:10.1134/S0001434614010295; Russian original Mat. Zametki 95 (2014), no. 2, 312--316, doi:10.4213/mzm10417.
ElPl21: Elsholtz, Christian and Planitzer, Stefan, Sums of four and more unit fractions and approximate parametrizations. Bull. Lond. Math. Soc. 53 (2021), no. 3, 695--709, doi:10.1112/blms.12452; arXiv:2012.05984v1 (2020).
El16: Elsholtz, C., Egyptian fractions with odd denominators. Q. J. Math. 67 (2016), no. 3, 425--430, doi:10.1093/qmath/haw020; arXiv:1606.02117v1 (2016).
DoTa25: van Doorn, W. and Tang, Q., The smallest denominator not contained in a unit fraction decomposition of $1$ with fixed length. arXiv:2512.22083 (v1 26 December 2025, v2 24 May 2026); Math. Proc. Cambridge Philos. Soc., published online 2026, doi:10.1017/S0305004126102102. Context: the link between $F(k)$ and the smallest missing denominator of Problem 293.
CFHMPSV25: Conlon, D., Fox, J., He, X., Mubayi, D., Pham, H. T., Suk, A. and Verstraëte, J., A question of Erdős and Graham on Egyptian fractions. Discrete Analysis 2025:28, doi:10.19086/da.154329. Context: a count with a different parameter.
Er50: Erdős, P., Az $1/x_1+\cdots+1/x_n=a/b$ egyenlet egész számú megoldásairól. Mat. Lapok 1 (1950), 192--210, p. 194. Context: the earliest statement of the counting question that the search located.

Problem 148

Status

Proof claims

Discussion