The Minimal Euclidean Function on the Gaussian Integers

Abstract

In 1949, Motzkin proved that every Euclidean domain R has a minimal Euclidean function, φ R . He showed that when R = Z, the minimal function is φ Z (x) = ⌊log 2 |x|⌋. For over seventy years, φ Z has been the only example of an explictly-computed minimal function in a number field. We give the first explicitly-computed minimal function in a non-trivial number field, φ Z[i] , which computes the length of the shortest possible (1 + i)-ary expansion of any Gaussian integer. We also present an algorithm that uses φ Z[i] to compute minimal (1 + i)-ary expansions of Gaussian integers. We solve these problems using only elementary methods.

An Elementary Proof of the Minimal Euclidean Function on the Gaussian Integers

“…We do this because Lenstra's proof requires an extensive knowledge of algebra, while this paper's arguments are elementary. As a consequence of Theorem 2.14 in [2] and Section 2.4, we answer Samuel's question by characterizing the sets φ −1…”

“…It then uses Lenstra's theorem (Equation 1) to show that φ Z[i] is given by that formula. [2]) Suppose that a+bi ∈ Z[i]\0, that 2 j a+bi, and that n is the least integer such that max a…”

“…The formula's proof in [2] provided a geometric description of the sets B n .Section 2.4 defines the geometry used in [2], and uses it to study our sets B n . Sections 2.4 and 3 then show that φ −1…”

“…A companion paper [2] gives a short proof of Theorem 1.2, using a result of Lenstra. Lenstra's proof requires comfort with a range of ideas in algebra.…”

“…We use our new geometric description of the sets B n to provide a shorter, alternative proof of Lenstra's theorem. This paper, therefore, provides a self-contained, elementary proof, at the expense of the brevity of [2].…”