  In this article, we are interested in situations where the existence of a
contiguous cascade of quantum resonant transitions is predicated on the
validity of a particular statement in number theory. The setting is a tailored
one-atom one-dimensional potential with a prescribed spectrum, under a weak
periodic perturbation. The former is, by now, an experimental reality [D.
Cassettari, G. Mussardo and A. Trombettoni, PNAS Nexus {\bf 2}, pgac279
(2022)].
  As a case study, we look at the following trivial statement: "Any power of
$3$ is an integer." Consequently, we "test" this statement in a numerical
experiment where we demonstrate an unimpeded upward mobility along an
equidistant, $\ln(3)$-spaced subsequence of the energy levels of a potential
with a log-natural spectrum, under a frequency $\ln(3)$ time-periodic
perturbation. We further show that when we "remove" $9$ from the set of
integers -- by excluding the corresponding energy level from the spectrum --
the cascade halts abruptly.
