The minimal Euclidean function on the Gaussian integers (Q2111255)

\textit{Th. Motzkin} [Bull. Am. Math. Soc. 55, 1142--1146 (1949; Zbl 0035.30302)], proved that every Euclidean domain \(R\) has a minimal Euclidean function, \(\phi_R\). He showed that when \(R = \mathbb{Z}\), the ring of integers, the minimal Euclidean function is \(\phi_{\mathbb{Z}(x)} = \lfloor \log_2 |x| \rfloor\), where \(\lfloor . \rfloor\), denotes the greatest integer function. For over seventy years, \(\phi _{\mathbb{Z}}\) has been the only example of an explicitly-computed minimal Euclidean function for the ring of integers of a number field. The author givesthe first explicitly-computed minimal Euclidean function in a non-trivial number field, the field of Gaussian integers, \(\phi_{\mathbb{Z}[i]}\), which also computes the length of the shortest possible \((1+ i)\)-ary expansion of any Gaussian integer. The author then presents an algorithm that uses \(\phi_{\mathbb{Z}[i]}\) to compute minimal \((1+ i)\)-ary expansions of Gaussian integers. He solves these problems using only elementary methods.