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Diophantine Equation

 Published on Friday, 26th March 2004, 06:00 pm; Solved by 21321;
Difficulty rating: 25%

Problem 66

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.