  The $N=\infty$ vector $O(N)$ model is a solvable, interacting field theory in
three dimensions ($D$). In a recent paper with A. Chubukov and J.
Ye~\cite{self}, we have computed a universal number, $\tilde{c}$,
characterizing the size dependence of the free energy at the
conformally-invariant critical point of this theory. The result~\cite{self} for
$\tilde{c}$ can be expressed in terms of polylogarithms. Here, we use
non-trivial polylogarithm identities to show that $\tilde{c}/N = 4/5$, a
rational number; this result is curiously parallel to recent work on
dilogarithm identities in $D=2$ conformal theories. The amplitude of the
stress-stress correlator of this theory, $c$ (which is the analog of the
central charge), is determined to be $c/N=3/4$, also rational. Unitary
conformal theories in $D=2$ always have $c = \tilde{c}$; thus such a result is
clearly not valid in $D=3$.
