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arXiv · 2609.28374

On the distribution of the minimal length of addition chains

Jean-Marie De Koninck·Nicolas Doyon·William Verreault
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Abstract

A sequence of integers $1=a_0 m^{0.9}$. Moreover, denoting by $G(m,r)$ the number of \emph{distinct} addition chains of length $m+r$ leading to an integer $n\in [2^m, 2^{m+1})$, we show that there exist positive constants $K_3$ and $K_4$ such that $$ K_3^r \left(\frac{m^2}{r}\right)^r\le G\left(m,r\right)\le K_4^r \left(\frac{m^2}{r} \right)^r $$ provided $m^{0.9}<r<m$. This improves and generalizes previous results on the minimal length of addition chains and addresses a question raised by Paul Erdős.

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Jean-Marie De Koninck, Nicolas Doyon, William Verreault. 2026-09-23. On the distribution of the minimal length of addition chains. https://arxiv.org/abs/2609.28374

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