In the course of doing some calculations I have found myself wanting to answer the following question:

Let $D/\mathbb{Q}$ be a quaternion algebra ramified at a prime $p$ and at $\infty$ and let $\mathcal{O}$ be a maximal order.

Is it possible to find $\lambda,\mu\in\mathcal{O}$ such that:

$1. N(\lambda) = p-1$

$2. N(\mu) = p$

$3. Tr(\lambda\overline{\mu})=0$

If the answer is no I would consider the weaker question of whether we are always able to choose $\mathcal{O}$ such that the above is possible.

I have definitely found examples of this occurrence for $p=2,3,5,7,11$ (cases where I have needed elements with these properties for certain applications) and can set congruence conditions on $p$ where it is possible etc, but I am wondering whether the result is true in general.

I have considered the idea of using the classification of such quaternion algebras and searching for obvious integral elements such as $\alpha + \beta i + \gamma j + \delta k$ with $\alpha,\beta,\gamma,\delta\in\mathbb{Z}$ with the required properties but it seems to be tough to do this way (even if we make simple choices for $\mu$ such as $\mu = j$ when the quaternion algebra is nicely presented).


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