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

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.