Consider quadratic Diophantine equations of the form: $$x^2 - Dy^2 = 1$$
For example, when $D=13$, the minimal solution in $x$ is $649^2 - 13 \times 180^2 = 1$.
It can be assumed that there are no solutions in positive integers when $D$ is square.
By finding minimal solutions in $x$ for $D = \{2, 3, 5, 6, 7\}$, we obtain the following:
$$\begin{align} 3^2 - 2 \times 2^2 &= 1\\ 2^2 - 3 \times 1^2 &= 1\\ {\color{red}{\mathbf 9}}^2 - 5 \times 4^2 &= 1\\ 5^2 - 6 \times 2^2 &= 1\\ 8^2 - 7 \times 3^2 &= 1 \end{align}$$Hence, by considering minimal solutions in $x$ for $D \le 7$, the largest $x$ is obtained when $D=5$.
Find the value of $D \le 1000$ in minimal solutions of $x$ for which the largest value of $x$ is obtained.
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