Transitions with the same energy difference in the Bohr model of the hydrogen atom

In the Bohr model of the hydrogen atom, the energy levels are a negative constant divided by the square of the level number. It is well known that special pairs of transitions exist that have the same energy difference, and a systematic treatment of this is given in the paper by Do and Phan (arXiv:2010.08338). In this paper we describe a simple method (using equal norms of Gaussian integers, and the Brahmagupta--Fibonacci identity) for finding all such pairs of transitions. We also analyze cascades of equal-frequency transitions, and use a theorem due to Fermat to show that cascades with more than three levels are not possible.