The exact Euclidean window of $\mathbb{Z}[\sqrt{14}]$

Source abstract

Let $R = \mathbb{Z}[\sqrt{14}]$, let $π= 4 + \sqrt{14}$, and let $\mathfrak{p} = (2,\sqrt{14}) = πR$ be the prime ideal above $2$. Cavallar and Lemmermeyer studied the multiplicative weighted norms $φ_{t}(x) = |N(x)|(t/2)^{v_π(x)}$ and asked for which weights $t$ they are Euclidean functions on $R$. We show that the set of these weights, the Euclidean window of $\mathfrak{p}$, is $(\sqrt{5},T]$, where $T$ is the algebraic number $$T = \sqrt{\frac{8}{104051\sqrt{2} - 55617\sqrt{7}}} = 2.2468286690\ldots = \sqrt{5} + 0.0107606915\ldots.$$ Thus the window is nonempty but not open. We also obtain the integer-valued completely multiplicative Euclidean function $2^{v_π(x)}|N(x)|^{6}$ on $R$.