Local association of index-prime orders in real quadratic fields

Let $p\in\mathbb N$ be prime, let $K=\mathbb Q[\sqrt{p}]$, and let $R_p$ be the order of index $p$ in $K$. An order is locally associated when it has the local association property defined in the paper. Local-association conjecture. The order $R_p$ is locally associated. Equivalently, $R_2$ is a locally associated order in $\mathbb Q[\sqrt{2}]$, and for every odd prime $p$, the Pell equation

$$x^2-y^2p=1$$

has an integer solution $(a,b)$ with $p\nmid b$. The claim gives a uniform criterion for local association of prime-index orders in real quadratic fields; the supplied text includes a theorem proving the corresponding equivalence, but does not establish the assertion for every prime, so its resolution should be checked.