  We consider the problem of determining whether a given prime p is a congruent
number. We present an easily computed criterion that allows us to conclude that
certain primes for which congruency was previously undecided, are in fact not
congruent. As a result, we get additional information on the possible sizes of
Tate-Shafarevich groups of the associated elliptic curves.
  We also present a related criterion for primes p such that 16 divides the
class number of the imaginary quadratic field Q(sqrt(-p)). Both results are
based on descent methods.
  While we cannot show for either criterion individually that there are
infinitely many primes that satisfy it nor that there are infinitely many that
do not, we do exploit a slight difference between the two to conclude that at
least one of the criteria is satisfied by infinitely many primes.
