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

TL;DR

This paper presents an elementary geometric proof for the minimal Euclidean function on Gaussian integers, simplifying previous algebra-heavy methods and answering an open question about its size distribution.

Contribution

It introduces a new geometric approach to prove the formula for the minimal Euclidean function on Gaussian integers without complex algebraic tools.

Findings

Derived an elementary proof for the minimal Euclidean function on Gaussian integers.

Provided a formula for the size of preimages of the minimal Euclidean function.

Included computational data and code for further analysis.

Abstract

Every Euclidean domain $R$ has a minimal Euclidean function, $\phi_R$. A companion paper \cite{Graves} introduced a formula to compute $\phi_{\mathbb{Z}[i]}$. It is the first formula for a minimal Euclidean function for the ring of integers of a non-trivial number field. It did so by studying the geometry of the set $B_n = \left \{ \sum_{j=0}^n v_j (1+i)^j : v_j \in \{0, \pm 1, \pm i \} \right \}$ and then applied Lenstra's result that $\phi_{\mathbb{Z}[i]}^{-1}([0,n]) = B_n$ to provide a short proof of $\phi_{\mathbb{Z}[i]}$. Lenstra's proof requires s substantial algebra background. This paper uses the new geometry of the sets $B_n$ to prove the formula for $\phi_{\mathbb{Z}[i]}$ without using Lenstra's result. The new geometric method lets us prove Lenstra's theorem using only elementary methods. We then apply the new formula to answer Pierre Samuel's open question: what is the size of $\phi_{\mathbb{Z}[i]}^{-1}(n)$?. Appendices provide a table of answers and the associated SAGE code.