Gaussian Integers and Arctangent Identities for Pi

utilize the x terms in Gregory’s series and have been instrumental in the calculation of decimal digits of π. Upon learning of these identities, one naturally desires an identity of the form π = r arctanx 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 note 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