Erdős Problem #293

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

For each fixed $k$, consider all increasing $k$-tuples of positive denominators with reciprocal sum one. The integer $v(k)$ is the least positive integer absent from their combined denominator sets, not the first missing denominator in just one expansion. [S1] [S2]

Statement recorded in the supplied source.

Let $k\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to $ 1=\frac{1}{n_1}+\cdots+\frac{1}{n_k} $ with $1\leq n_1<\cdots <n_k$. Estimate the growth of $v(k)$. [S1]

Short English statement

How quickly does the smallest denominator absent from every fixed-length distinct-unit-fraction expansion of one grow with the allowed length?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2034 (EP-293), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 293, https://www.erdosproblems.com/293. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Wouter van Doorn and Quanyu Tang, arXiv:2512.22083 (2025). https://arxiv.org/abs/2512.22083. Bibliographic information from the supplied record.

• S4 Bleicher, M. N. and Erdős, P., The number of distinct subsums of $\sum \sb{1}^{N}\,1/i$. Math. Comp. (1975), 29-42. Bibliographic information from the supplied record.

• S5 W. van Doorn and Q. Tang, The smallest denominator not contained in a unit fraction decomposition of 1 with fixed length. arXiv:2512.22083 (2025). Bibliographic information from the supplied record.