---

Problem 420

---

2x2 positive integer matrix

A positive integer matrix is a matrix whose elements are all positive integers.
Some positive integer matrices can be expressed as a square of a positive integer matrix in two different ways. Here is an example:

We define F(N) as the number of the 2x2 positive integer matrices which have a trace less than N and which can be expressed as a square of a positive integer matrix in two different ways.
We can verify that F(50) = 7 and F(1000) = 1019.

Find F(10⁷).

---

2x2正整数矩阵

正整数矩阵是元素均为正整数的矩阵。
有些正整数矩阵可以有多种方式表达为另一个正整数矩阵的平方，下面是一个例子：

定义F(N)为迹小于N，且能够表示成两个不同正整数矩阵平方的正整数矩阵数目。
可以验证F(50) = 7以及F(1000) = 1019。

求F(10⁷)。

---