  Non-linear recurrences which generate integers in a surprising way have been
studied by many people. Typically people study recurrences that are linear in
the highest order term. In this paper I consider what happens when the
recurrence is not linear in the highest order term. In this case we no longer
produce a unique sequence, but we sometimes have surprising results. If the
highest order term is raised to the $m^{th}$ power we expect answers to have
$m^{th}$ roots, but for some specific recurrences it happens that we generate
rational numbers ad infinitum. I will give a general example in the case of a
first order recurrence with $m=2$, and a more specific example that is order 3
with $m=2$ which comes from a generalized Somos recurrence.
