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2x2 positive integer matrix

Problem 420

Published on Sunday, 24th March 2013, 01:00 am; Solved by 238

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).