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We study the distribution of Hasse's unit index $Q(L)$ for the CM-fields $L = \mathbb{Q}(\sqrt{d}, \sqrt{-1})$ as $d$ varies among positive squarefree integers. We prove that the number of $d\leq X$ such that $Q(L) = 2$ is proportional to $X/\sqrt{\log X}$., 11 pages

Let $E\in\{-1, \pm 2\}$. We improve on the upper and lower densities of primes $p$ such that the equation $x^2-2py^2=E$ is solvable for $x, y\in \mathbb{Z}$. We prove that the natural density of primes $p$ such that the narrow class group of the real quadratic number field $\mathbb{Q}(\sqrt{2p})$ has an element of order $16$ is equal to $\frac{1}{64}$. We give an application of our results to the distribution of Hasse's unit index for the CM-fields $\mathbb{Q}(\sqrt{2p}, \sqrt{-1})$. Our results are consequences of a twisted joint distribution result for the $16$-ranks of class groups of $\mathbb{Q}(\sqrt{-p})$ and $\mathbb{Q}(\sqrt{-2p})$ as $p$ varies., 15 pages