In this paper we study transitions of atoms between energy levels of several number-theory-inspired trapping potentials under the effect of time-dependent perturbations. First, we simulate in detail the case of a trap whose single-particle spectrum is given by the prime numbers. We investigate one- body Rabi oscillations and the excitation lineshape for two resonantly coupled energy levels, and we show that quantum control is a faster method for state preparation than periodic perturbation. Next, we investigate cascades of such transitions, particularly whether one can construct a quantum system where the existence of a continuous resonant cascade from a given initial energy eigenstate is predicated by the validity of a given statement in number theory. We find that such resonance cascades, in a suitably-designed one-body system, can be used to verify that the sequence of natural numbers is closed under multiplication. We further present ideas for two more resonance cascade experiments designed to illustrate the validity of the Diophantus-Brahmagupta-Fibonacci identity and the validity of the Goldbach conjecture.
