The four-squares conjecture for generalized Pell sequences

Let $a$, $b$ and $d$ be positive integers such that $d$ is not a square, let $\alpha=a+b^{2}\sqrt{d}$ have norm

$$N_{\alpha}=a^{2}-b^{4}d,$$

and let $\varepsilon=(t+u\sqrt{d})/2$ be a unit in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ with $t$ and $u$ positive integers. Define integer sequences $(x_k)_{k=-\infty}^{\infty}$ and $(y_k)_{k=-\infty}^{\infty}$ by

$$x_k+y_k\sqrt{d}=\alpha\varepsilon^{2k}.$$

Assume that $b^2=y_0$ is the smallest square among the $y_k$ and let $\operatorname{sf}(n)$ denote the unique squarefree integer such that $n/\operatorname{sf}(n)$ is a square. Four-squares conjecture. There are at most four distinct integer squares among the $y_k$. If $\operatorname{sf}(|N_{\alpha}|)\mid 2p$ for an odd prime $p$, then there are at most three distinct integer squares among the $y_k$. Furthermore, if $|N_{\alpha}|$ is a perfect square, then there are at most two distinct integer squares among the $y_k$. These sequences give solutions to the generalized Pell equation $x^2-dy^2=N_{\alpha}$, and the conjecture predicts strong bounds on square values in such sequences; the sharper bounds depend on the arithmetic of the norm $N_{\alpha}$.