Every Pell equation has a positive nontrivial integral solution

Let $\mathrm{\ell }$ be the period length from [F1]. Applying [F2] at $n=\mathrm{\ell }-1$ and using $Q_{\mathrm{\ell }}=1$ gives $p_{\mathrm{\ell }-1}^2-D{q}_{\mathrm{\ell }-1}^2=(-1{)}^{\mathrm{\ell }}\mathrm{.}$

If $\mathrm{\ell }$ is even, step 1.1 already gives a norm-one solution. Since $q_{\mathrm{\ell }-1}>0$ for every convergent denominator and $p_{\mathrm{\ell }-1}^2=D{q}_{\mathrm{\ell }-1}^2+1$, one has $p_{\mathrm{\ell }-1}>0$ and $q_{\mathrm{\ell }-1}>0$, so $(p_{\mathrm{\ell }-1},q_{\mathrm{\ell }-1})$ is a positive nontrivial solution.

If $\mathrm{\ell }$ is odd, step 1.1 gives $p_{\mathrm{\ell }-1}^2-D{q}_{\mathrm{\ell }-1}^2=-1.$ Put $\alpha :=p_{\mathrm{\ell }-1}+q_{\mathrm{\ell }-1}\sqrt{D}\mathrm{.}$ Then $p_{\mathrm{\ell }-1}>0$ and $q_{\mathrm{\ell }-1}>0$ as in step 2.1, and [F3] gives $N_D({\alpha }^2)=N_D(\alpha {)}^2=(-1{)}^2=1.$ Explicitly, ${\alpha }^2=(p_{\mathrm{\ell }-1}^2+D{q}_{\mathrm{\ell }-1}^2)+2p_{\mathrm{\ell }-1}{q}_{\mathrm{\ell }-1}\sqrt{D},$ so ${\alpha }^2$ gives a positive nontrivial integral solution of Pell's equation.

Either step 2.1 or step 3.1 supplies the required positive nontrivial solution.