  We generalize the classical theory of periodic continued fractions (PCFs)
over ${\mathbf Z}$ to rings ${\mathcal O}$ of $S$-integers in a number field.
Let ${\mathcal B}=\{\beta, {\beta^*}\}$ be the multi-set of roots of a
quadratic polynomial in ${\mathcal O}[x]$. We show that PCFs
$P=[b_1,\ldots,b_N,\bar{a_1\ldots ,a_k}]$ of type $(N,k)$ potentially
converging to a limit in ${\mathcal B}$ are given by ${\mathcal O}$-points on
an affine variety $V:=V({\mathcal B})_{N,k}$ generically of dimension $N+k-2$.
We give the equations of $V$ in terms of the continuant polynomials of Wallis
and Euler. The integral points $V({\mathcal O})$ are related to writing
matrices in $\textrm{SL}_2({\mathcal O})$ as products of elementary matrices.
We give an algorithm to determine if a PCF converges and, if so, to compute its
limit.
  Our standard example generalizes the PCF $\sqrt{2}=[1,\bar{2}]$ to the
${\mathbf Z}_2$-extension of ${\mathbf Q}$: $F_n={\mathbf Q}(\alpha_n)$,
$\alpha_{n}:=2\cos(2\pi/2^{n+2})$, with integers ${\mathcal O}_n={\mathbf
Z}[\alpha_n]$. We want to find the PCFs of $\alpha_{n+1}$ over ${\mathcal
O}_{n}$ of type $(N,k)$ by finding the ${\mathcal O}_{n}$-points on
$V({\mathcal B}_{n+1})_{N,k}$ for ${\mathcal B}_{n+1}:=\{\alpha_{n+1},
-\alpha_{n+1}\}$. There are three types $(N,k)=(0,3), (1,2), (2,1)$ such that
the associated PCF variety $V({\mathcal B})_{N,k}$ is a curve; we analyze these
curves. For generic ${\mathcal B}$, Siegel's theorem implies that each of these
three $V({\mathcal B})_{N,k}({\mathcal O})$ is finite. We find all the
${\mathcal O}_n$-points on these PCF curves $V({\mathcal B}_{n+1})_{N,k}$ for
$n=0,1$. When $n=1$ we make extensive use of Skolem's $p$-adic method for
$p=2$, including its application to Ljunggren's equation $x^2 + 1 =2y^4$.

Number Theory;Algebraic Geometry