DescriptionThe problem of explicitly computing eigenvalues of the Dirichlet Laplacian on two dimensional domains is in general very difficult.However, for certain highly symmetric domains it is possible to obtain closed-form expressions for eigenvalues. In the case of rectangular domains not only does such a closed-form expression exist, but it also happens to coincide with the formula for integer points on an ellipse. Establishing criteria for the existence of such points as well as determining their multiplicity was a major driving force in the development of 18th and 19th century number theory, leading to the Law of Quadratic Reciprocity, the Hilbert-Waring theorem, the Gauss circle problem, and many other theorems and conjectures. The goal of this thesis is to use some of these classical number-theoretic results to answer a question about multiplicities of Dirichlet eigenvalues on certain special rectangles. Specifically, we show that on these rectangles the set of multiplicities of Dirichlet eigenvalues is equal to the whole of N.