Home / Journals / Uniform Distribution Theory (UDT) / UDT. Volume 21. Issue 1 / Divergent trajectories in the $3⁒π‘₯ +1$ problem and a $𝑆 βˆ’π‘ˆβ’π‘›β’π‘–β’π‘‘β’π‘ $ Equation

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Journal

Uniform Distribution Theory (UDT)


Volume & Issue

UDT. Volume 21. Issue 1


Published on

June, 2026


Pages

1-9


DOI

https://doi.org/10.66833/UDT-2026-0001

Article

Divergent trajectories in the $3⁒π‘₯ +1$ problem and a $𝑆 βˆ’π‘ˆβ’π‘›β’π‘–β’π‘‘β’π‘ $ Equation


Authors

Jean-Louis Verger-Gaugry


Abstract

In this note we assume the existence of divergent trajectories (if any) under the Collatz map and investigate the consequences of this assumption, even though the Conjecture of Collatz seems to be true. In a trajectory we show that the Diophantine exponential equation $x+y=w2^{n}$ in the integers $x,y,w,n$ plays an important role. In this context the recent strong finiteness theorem of Bennett and Billerey on $𝑆-units$ has consequences on the set of prime numbers generated by a collection of local minima, called wave-minima (defined ad hoc for simplicity’s sake) in a divergent trajectory. It allows to define canonically a density of a trajectory and some conjectures and open problems related to these prime numbers.


Citation

Verger-Gaugry, JL. (2026) Divergent trajectories in the 3x+1 problem and a S-Units Equation. Uniform Distribution Theory. 21(1) 1-9. https://doi.org/10.66833/UDT-2026-0001