Consider a right triangle of integral lengths, like 3-4-5. It is a very ancient problem as to whether an integer, when squared, can be represented as the sum of two squared nonzero integers, hence producing such a triangle. The goal of this talk will be to present the surprisingly elegant and simple proofs of the two and four- squares theorems using techniques from complex analysis and the Jacobi theta function. Although elementary proofs exist for these theorems, it would be beneficial to get a flavor for the form of this proof, as this method can be readily generalized to the theory of Shimura curves in the study of automorphic forms.
This talk should be accessible to everyone -- elementary analysis recommended.