Problem 294

Let $N\geq 1$ and let $t(N)$ be the least integer $t$ such that there is no solution to

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

with $t=n_1<\cdots <n_k\leq N$ . Estimate $t(N)$ .

Source: erdosproblems.com/294

Number theory
Unit fractions

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An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

PROVED, the site's label, which attaches to the estimate: Liu and Sawhney (Int. Math. Res. Not. 2026) determine $t(N)$ up to a factor $(\log\log N)^3(\log\log\log N)^{O(1)}$ , namely $N/((\log N)(\log\log N)^3(\log\log\log N)^{O(1)})\ll t(N)\ll N/\log N$ , the upper bound being the one Erdős and Graham had stated. This is the accepted claim Liu and Sawhney 2024, refereed and credited by the site's curator. The derived standing is solved, answered rather than proved, because the question asks for an estimate rather than a yes or no. The exact order of $t(N)$ inside that window is not 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), printed p. 35. Library home: Erdős and Graham 1980.
LiSa24: Liu, Y. P. and Sawhney, M., On further questions regarding unit fractions. arXiv:2404.07113v1 (10 April 2024); Int. Math. Res. Not. 2026, no. 2, rnaf382, DOI 10.1093/imrn/rnaf382, published online 14 January 2026. Theorem 1.6. Library home: Liu and Sawhney 2024.

Problem 294

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