0%

Problem 66


Problem 66


Diophantine equation

Consider quadratic Diophantine equations of the form:

x2 – Dy2 = 1

For example, when D=13, the minimal solution in x is 6492 – 13×1802 = 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:

32 – 2×22 = 1
22 – 3×12 = 1
92 – 5×42 = 1
52 – 6×22 = 1
82 – 7×32 = 1

Hence, by considering minimal solutions in x for D ≤ 7, the largest x is obtained when D=5.

Find the value of D ≤ 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:

x2 – Dy2 = 1

举例而言,当D=13时,x的最小值出现在6492 – 13×1802 = 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}分别求出x取最小值的解,我们得到:

32 – 2×22 = 1
22 – 3×12 = 1
92 – 5×42 = 1
52 – 6×22 = 1
82 – 7×32 = 1

因此,对于所有D ≤ 7,当D=5时x的最小值最大。

对于D ≤ 1000,求使得x的最小值最大的D值。