A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163

In this paper, we give an elementary proof of the fact that the rings are unique factorization domains for the values d = 3, 7, 11, 19, 43, 67, 163. While the result in itself is well known, our proof is new and completely elementary and uses neither the Minkowski convex body theorem, nor the Dedekind and Hasse theorems. Furthermore, it does not use either the theory of algebraic integers, or the theory of Noetherian rings. It only uses basic notions from the theory of commutative rings.