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.
La représentation des nombres dans la base rationnelle $p/q$ a été introduite en 2008 par Akiyama, Frougny & Sakarovitch, avec un accent particulier sur le cas $p/q=3/2$. Des questions naturelles liées aux représentations dans cette base spécifique, passées inaperçues jusqu’ici, s’avèrent impliquer intimement la fonction de Collatz $3x+1$. Le but de cette note est d’exposer ces liens et de motiver des recherches plus approfondies sur ceux-ci.
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Keywords: Collatz conjecture, iteration, numeration, odometer
Mots-clés : Conjecture de Collatz, itération, numération, odomètre
Shalom Eliahou 1, 2; Jean-Louis Verger-Gaugry 3
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@article{CRMATH_2025__363_G4_329_0,
author = {Shalom Eliahou and Jean-Louis Verger-Gaugry},
title = {The number system in rational base~$3/2$ and the $3x+1$ problem},
journal = {Comptes Rendus. Math\'ematique},
pages = {329--336},
publisher = {Acad\'emie des sciences, Paris},
volume = {363},
year = {2025},
doi = {10.5802/crmath.662},
language = {en},
}
TY - JOUR AU - Shalom Eliahou AU - Jean-Louis Verger-Gaugry TI - The number system in rational base $3/2$ and the $3x+1$ problem JO - Comptes Rendus. Mathématique PY - 2025 SP - 329 EP - 336 VL - 363 PB - Académie des sciences, Paris DO - 10.5802/crmath.662 LA - en ID - CRMATH_2025__363_G4_329_0 ER -
Shalom Eliahou; Jean-Louis Verger-Gaugry. The number system in rational base $3/2$ and the $3x+1$ problem. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 329-336. doi : 10.5802/crmath.662. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.662/
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