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arXiv · 2504.13716

The number system in rational base $3/2$ and the $3x+1$ problem

Shalom Eliahou·Jean-Louis Verger-Gaugry
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Abstract

The representation of numbers in rational base $p/q$ was introduced in 2008 by Akiyama, Frougny & Sakarovitch, with a special focus on the case $p/q=3/2$. Unnoticed since then, natural questions related to representations in that specific base turn out to intimately involve the Collatz $3x+1$ function. Our purpose in this note is to expose these links and motivate further research into them.

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Shalom Eliahou, Jean-Louis Verger-Gaugry. 2025-04-18. The number system in rational base $3/2$ and the $3x+1$ problem. https://arxiv.org/abs/2504.13716

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