A number theoretic result for Berge's conjecture

(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 $p > k^2$.