  A rational projective plane ($\mathbb{QP}^2$) is a simply connected, smooth,
closed manifold $M$ such that $H^*(M;\mathbb{Q}) \cong
\mathbb{Q}[\alpha]/\langle \alpha^3 \rangle$. An open problem is to classify
the dimensions at which such a manifold exists. The Barge-Sullivan rational
surgery realization theorem provides necessary and sufficient conditions that
include the Hattori-Stong integrality conditions on the Pontryagin numbers. In
this article, we simplify these conditions and combine them with the signature
equation to give a single quadratic residue equation that determines whether a
given dimension supports a $\mathbb{QP}^2$. We then confirm existence of a
$\mathbb{QP}^2$ in two new dimensions and prove several non-existence results
using factorizations of numerators of divided Bernoulli numbers. We also
resolve the existence question in the Spin case, and we discuss existence
results for the more general class of rational projective spaces.
