  We investigate Hamiltonians with attractive interactions between pairs of
fermions coupled to angular momentum J. We show that pairs with spin J are
reasonable building blocks for the low-lying states. For systems with only a J
= Jmax pairing interaction, eigenvalues are found to be approximately integers
for a large array of states, in particular for those with total angular momenta
I le 2j. For I=0 eigenstates of four fermions in a single-j shell we show that
there is only one non-zero eigenvalue. We address these observations using the
nucleon pair approximation of the shell model and relate our results with a
number of currently interesting problems.
