Problem 66

Diophantine Equation

Consider quadratic Diophantine equations of the form:

$x^{2} - D y^{2} = 1$

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:

$\begin{aligned} 3^{2} - 2 \times 2^{2} & = 1 \\ 2^{2} - 3 \times 1^{2} & = 1 \\ 9^{2} - 5 \times 4^{2} & = 1 \\ 5^{2} - 6 \times 2^{2} & = 1 \\ 8^{2} - 7 \times 3^{2} & = 1 \end{aligned}$

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.

丢番图方程

考虑如下形式的二次丢番图方程：

$x^{2} - D y^{2} = 1$

举例而言，当 $D = 13$ 时， $x$ 的最小值出现在 $649^{2} - 13 \times 180^{2} = 1$ 。

可以断定，当 $D$ 是平方数时，这个方程不存在正整数解。

对于 $D = {2, 3, 5, 6, 7}$ ， $x$ 取最小值的解分别是：

$\begin{aligned} 3^{2} - 2 \times 2^{2} & = 1 \\ 2^{2} - 3 \times 1^{2} & = 1 \\ 9^{2} - 5 \times 4^{2} & = 1 \\ 5^{2} - 6 \times 2^{2} & = 1 \\ 8^{2} - 7 \times 3^{2} & = 1 \end{aligned}$

因此，对于所有 $D \leq 7$ ，当 $D = 5$ 时 $x$ 的最小值最大。

对于所有 $D \leq 1000$ ，求使得 $x$ 的最小值最大时 $D$ 的取值。