I am confused by:

Thus, setting $a = m^2, b = 2mn, c= n^2$ and $d=(m+n)^2$ gives us a Pythagorean quadruple.

I think it should actually be

Thus, setting $a^2 = m^2, b^2 = 2mn, c^2= n^2$ and $d^2=(m+n)^2$ gives us a Pythagorean quadruple.

It should be noted that $b^2= 2 m n$ does not give us an integer $b$ unless $2 m n$ is a perfect square. In the following set of equations $ d=m+n$ and not $d=(m+n)^2$.

The section on extending to longer sequences is very interesting.


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