  (Original version of PhD thesis, submitted in Spring 2009 to Harvard
University. Provides a solution of the $p > k^2$ case, corresponding to Berge
families I-VI, of the "Lens space realization problem" later solved in entirety
by Greene.) In the 1980's, Berge proved that a certain collection of knots in
$S^3$ admitted lens space surgeries, a list which Gordon conjectured was
exhaustive. More recently, J. Rasmussen used techniques from Heegaard Floer
homology to translate the related problem of classifying simple knots in lens
spaces admitting L-space homology sphere surgeries into a combinatorial number
theory question about the data $(p,q,k)$ associated to a knot of homology class
$k \in H_1(L(p,q))$ in the lens space $L(p,q)$. In the following paper, we
solve this number theoretic problem in the case of $p > k^2$.
