Problem Archive

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 $2\times 2$ positive integer matrices which have a tracethe sum of the elements on the main diagonal 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 $F(10^7)$.