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2x2 positive integer matrix

Published on Sunday, 24th March 2013, 01:00 am; Solved by 347;
Difficulty rating: 60%

Problem 420

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).