  Kaplansky [2003] proved a theorem on the simultaneous representation of a
prime $p$ by two different principal binary quadratic forms. Later, Brink found
five more like theorems and claimed that there were no others. By putting
Kaplansky-like theorems into the context of threefield identities after
Andrews, Dyson, and Hickerson, we find that there are at least two similar
results not on Brink's list. We also show how such theorems are related to
results of Muskat on binary quadratic forms
