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arXiv · 2512.22083

The smallest denominator not contained in a unit fraction decomposition of $1$ with fixed length

Wouter van Doorn·Quanyu Tang
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Abstract

Let $v(k)$ be the smallest integer larger than $1$ that does not occur among the denominators in any identity of the form $$ 1=\frac1{n_1}+\cdots+\frac1{n_k}, $$ where $1 \le n_1<\cdots 0$ such that the inequality $$ v(k)\ge e^{c k^2} $$ holds for all positive integers $k$.

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Wouter van Doorn, Quanyu Tang. 2025-12-26. The smallest denominator not contained in a unit fraction decomposition of $1$ with fixed length. https://doi.org/10.1017/s0305004126102102

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