  Is it possible that two different transitions in the non-relativistic quantum
mechanical model of the hydrogen atom give the same frequency? That is, can
different energy level transitions in a hydrogen atom have the same photon
radiation frequency? This question, which was asked during a Ph.D. oral exam in
1997 at the University of Colorado Boulder, is well-known among physics
graduate students. We show a general solution to this question, in which all
equifrequency transition pairs can be obtained from the set of solutions of a
Diophantine equation. This fun puzzle is a simple yet concrete example of how
number theory can be relevant to quantum systems, a curious theme that emerges
in theoretical physics but is usually inaccessible to the general audience.
