Peer-reviewedCC BY-NC-ND 4.0Uniform Distribution Theory
Divergent Trajectories in the 3x+1 Problem and a em S- Units Equation
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 = w 2n in the integers $x, y, w, n$ plays an important role. In this context the recent strong finiteness theorem of Bennett and Billerey on ${\it 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.
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BibTeX
@article{vergergaugry2026divergent,
title = {{Divergent Trajectories in the $3x+1$ Problem and a $\em S$- Units Equation}},
author = {Jean-Louis Verger-Gaugry},
journal = {Uniform Distribution Theory},
year = {2026},
month = oct,
doi = {10.66833/udt-2026-0001},
url = {https://hal.science/hal-05857708v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - JOUR
TI - Divergent Trajectories in the 3x+1 Problem and a em S- Units Equation
AU - Jean-Louis Verger-Gaugry
PY - 2026
DA - 2026/10/05
JO - Uniform Distribution Theory
DO - 10.66833/udt-2026-0001
UR - https://hal.science/hal-05857708v1
AB - 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 = w 2^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 ${\it 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.
SN - 2309-5377
SN - 1336-913X
ER -