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In the classical theory, a famous by-product of the continued fractionexpansion of quadratic irrational numbers $\sqrt{D}$ is the solution to Pell'sequation for $D$. It is well-known that, once an integer solution to Pell'sequation exists, we can use it to generate all other solutions$(u_n,v_n)_{n\in\Zee}$. Our object of interest is the polynomial version ofPell's equation, where the integers are replaced by polynomials with complexcoefficients. We then investigate the factors of $v_n(t)$. In particular, weshow that over the complex polynomials, there are only finitely many values of$n$ for which $v_n(t)$ has a repeated root. Restricting our analysis to$\Qee[t]$, we give an upper bound on the number of "new" factors of $v_n(t)$ ofdegree at most $N$. Furthermore, we show that all "new" linear rational factorsof $v_n(t)$ can be found when $n\leq 3$, and all "new" quadratic rationalfactors when $n\leq 6$.

Properties of solutions to Pell's equation over the polynomial ring