arXiv:2611.14569v1 Announce Type: new 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 $x + y = w 2^n$ 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.
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