Diophantine equation

Consider quadratic Diophantine equations of the form:

\[ x^{2} – Dy^{2} = 1 \]

For example, when D\( = \)1\( 3 \), the minimal solution in \( x \) is \( 6492 - 13 \times 1802=1 \).

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:

\[ 3^{2} – 2×2^{2} = 1\\\\ 2^{2} – 3×1^{2} = 1\\\\ \color{red}{9}^{2} – 5×4^{2} = 1\\\\ 5^{2} – 6×2^{2} = 1\\\\ 8^{2} – 7×3^{2} = 1 \]

Hence, by considering minimal solutions in \( x \) for \( D \leq 7 \), the largest \( x \) is obtained when \( D = 5 \).

Find the value of \( D \leq 1000 \) in minimal solutions of \( x \) for which the largest value of \( x \) is obtained.