Gaussian Integers and Arctangent Identities for π

π = r arctan x where r and x are rational and |x | < 1 is small. Such an identity would require only one evaluation of the arctangent function and this evaluation would converge quickly. However, identities of this form do not exist and this fact is not mentioned in the literature alongside lists of such multiple-angle identities. The present article gives a very natural proof of this fact using a simple consequence of unique factorization of Gaussian integers (Main Lemma, Section 2). Section 3 gives several applications of the Main Lemma to arctangent identities, triangles, polygons on geoboards, and