Show that if $p_k/q_k$ is a convergent of the simple continued fraction expansion of $\sqrt{d}$, then $|p^2_k − dq^2_k | < 1 + 2\sqrt{d}$.

Question

I believe this comes from the Pell Equation and stems from $|\frac{p}{q}-\sqrt{d}| <\frac{1}{2q^2}$ but I'm not completely sure.

Answer 1

You only have that $\left|\frac{p_k}{q_k}-\sqrt{d}\right|<\frac{1}{q_k^2}$, in general - $\frac{1}{2q_k^2}$ is true for some, but not all, $k$.

Now write:

$$p_k^2-dq_k^2=q_k^2\left(\frac{p_k}{q_k}+\sqrt{d}\right)\left(\frac{p_k}{q_k}-\sqrt{d}\right)$$

and compute upper bounds for the terms.