### Parity Considerations in the Expansion of Fermat-Pell Polynomials, by J. Mc Laughlin

For each positive integer $n$ it is shown how to construct a finite collection of
multivariable polynomials
$\{F_{i}:=F_{i}(t,X_{1},\cdots, X_{\lfloor \frac{n+1}{2} \rfloor})\}$
such that each positive integer whose squareroot has a continued fraction
expansion with period $n+1$ lies in the range of exactly one of these polynomials.
Moreover, each of these polynomials
satisfy a polynomial Pell's equation
$C_{i}^{2} -F_{i}H_{i}^{2} = (-1)^{n-1}$
(where $C_{i}$ and $H_{i}$ are polynomials in the variables
$t,X_{1},\cdots, X_{\lfloor \frac{n+1}{2} \rfloor}$)
and the fundamental solution can be written down. Likewise, if all the
$X_{i}$'s and $t$ are non-negative
then the continued fraction
expansion of
$\sqrt{F_{i}}$ can
be written down.
Furthermore,
the congruence class modulo 4 of
$F_{i}$
depends in a simple way on the variables
$t,X_{1},\cdots, X_{\lfloor \frac{n+1}{2} \rfloor}$
so that the fundamental unit can be written down
for a large class of real quadratic fields.
Along the way a complete solution is
given to the problem of determining for which symmetric strings of positive integers
$a_{1},\cdots, a_{n}$
do there exist positive integers $D$ and $a_{0}$ such that
$\sqrt{D} = [\,a_{0};\overline{a_{1} , ..., a_{n},2a_{0}}]$.

J. Mc Laughlin <jgmclaug@math.uiuc.edu>