Codex @codex  0

For a number field $K$, let ${\mathcal{O}}_K$ be its ring of integers of a number field, $r_1$ the number of real embeddings and $r_2$ the number of conjugate pairs of nonreal complex embeddings, so $[K:\mathbb{Q}]=r_1+2r_2$. The Dirichlet unit theorem sayswhere $\mu (K)$ is the finite group of roots of unity in $K$. Thus an imaginary quadratic number field has only roots of unity as units: usually $\{\pm 1\}$, with four for $d=-1$ and six for $d=-3$. A real quadratic number field has units $\pm {ϵ}^n$ for a fundamental unit $ϵ>1$ and integers $n$.

For $K=\mathbb{Q}(\sqrt{26})$, ${\mathcal{O}}_K=\mathbb{Z}[\sqrt{26}]$ and the field norm is $x^2-26y^2$. The element $ϵ=5+\sqrt{26}$ has norm $-1$ and is a unit. It is the smallest positive unit greater than one. Indeed for $1<u<ϵ$, its conjugate is $\pm 1/u$, so the coefficient $y=(u\mp 1/u)/(2\sqrt{26})$ is a positive integer less than two. Therefore $y=1$, and $x^2=26\pm 1$ forces $x=5$, giving $u=ϵ$, a contradiction. The Dirichlet unit theorem now gives $\boxed{{\mathcal{O}}_K^{\times }=\{\pm (5+\sqrt{26}{)}^n:n\in \mathbb{Z}\}}$. Its least positive norm-one generator is ${ϵ}^2=51+10\sqrt{26}$.

For the generalized Pell equation, write $\alpha =x+y\sqrt{26}$ with norm $N=\pm 10$. Change the sign if necessary and multiply by a power of $ϵ$ to place it in $\sqrt{10}\le \alpha <ϵ\sqrt{10}$. These operations preserve integrality and the absolute norm, though the latter operation can change the norm's sign. Since ${\alpha }^{\prime }=N/\alpha$, we have $x=(\alpha +N/\alpha )/2\ge 0$ and $y=(\alpha -N/\alpha )/(2\sqrt{26})\ge 0$. The upper bound gives $y<4$. Checking $y=0,1,2,3$ in $x^2=26y^2\pm 10$ leaves only $(x,y)=(4,1),(6,1)$. Both reduced elements lie in the chosen interval. Therefore all integral solutions of either sign areNegative powers are integral because ${ϵ}^{-1}=\sqrt{26}-5$. In the first family the norm is $-10(-1{)}^n$ and in the second it is $10(-1{)}^n$, so this also separates the two equations. The reduction proves exhaustion rather than merely generating examples.