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:

$$\begin{pmatrix} 40 & 12\ 48 & 40 \end{pmatrix} = \begin{pmatrix} 2 & 3\ 12 & 2 \end{pmatrix}^2 = \begin{pmatrix} 6 & 1\ 4 & 6 \end{pmatrix}^2 $$

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).

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter