A number theoretic result for Berge's Conjecture

In [2], Berge offered a conjectured classification of simple knots in S 3 admitting lens space surgeries. More recently [6], 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 number theoretic question about the data ( p, q, k ) associated to a knot of homology class k ∈ H 1 ( L ( p, q )) in the lens space L ( p, q ). He conjectured that the solution to this number theoretic problem should prove Berge's conjectured classification of simple knots in S 3 admitting lens space surgeries. In the following paper, we solve this number theoretic problem in the case of p > k 2 .