Gerry, then let's just close it.

Interesting typo in lawsofform.org/gsb/nature.html : "the difficulty of the foul colour problem" :-)

So this was a homework question?

I'd say that physics was pretty much an application of eigenvalues and eigenvectors. :-) In particular normal modes (en.wikipedia.org/wiki/Normal_modes) of oscillations for a system with $n$ degrees of freedom comes down to finding eigenvalues/vectors of an $n$-by-$n$ matrix.

The product notation $\prod_{n=1}^{2^{n^2}}$ is rather treacherous when dealing with a noncommutative ring. There are orderings of your $M_{(n,k)}$ having two consecutive matrices which multiply to zero.

Also posted at math.stackexchange.com/questions/5216/…

Schoof's book reprints Euler's argument in the original Latin :-)

What about it? What actually is it? To me it doesn't appear fairly natural, but rather daft.

This one is a real problem :-)