Whether the U($2N$) symmetry of Dirac fermions in 2+1 space-time dimensions is spontaneously broken by pair condensation once interactions are present is an important problem in non-perturbative quantum field theory. Here I focus on the Thirring model, whose interaction is a current-current contact term, using numerical simulations of a lattice model formulated with domain wall fermions -- it has previously been demonstrated that U($2N$) symmetry is recovered in the limit of infinite wall separation. I present results obtained with flavor numbers $N=1$ and 2, and will attempt to put both upper and lower bounds on $N_c$, the critical number of flavors above which symmetry breaking does not occur even for arbitrarily strong coupling. The resulting $N_c$ will be shown to be very far from the value $N_c\approx6.6$ obtained using staggered lattice fermions, which do not observe U($2N$) symmetry.