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(10^{7}).

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