# A product of two sums of four squares

I am dealing with a problem and I hope you can help me. I have already proved this:

Let us suppose that integers $m$ and $n$ can be written as sum of squares of two integers. Prove that m*n can also be written as sum of squares of two integers.

Now I am trying to prove this: Let us suppose that integers $m$ and $n$ can be written as the sum of squares of four integers, i.e., $m=m_1^2+m_2^2+m_3^2+m_4^2$ and $n=n_1^2+n_2^2+n_3^2+n_4^2$. Prove that $mn$ can also be written as sum of squares of four integers.

Can you help me on this? Thank You for reading my post.

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Please see Euler's four squares identity.

It is analogous to the identity that you probably used to prove the result about the product of two sums of two squares. That identity can be interpreted as a result about the norm of the product of two complex numbers.

And it turns out that the four-squares identity can be interpreted as a result about the norm of a product of two quaternions. But Euler did not know that, since quaternions were only discovered many years later.

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lol of course Euler solved this one already :-P – timidpueo Mar 23 '13 at 18:54
But he did not manage to show that every non-negative integer is a sum of four squares. Lagrange did that. – André Nicolas Mar 23 '13 at 18:56
ah icic... well I guess it's good to know that there were still some things left that Euler didn't prove... – timidpueo Mar 23 '13 at 19:02

Just like the sum of two squares is related to the norm (square of the absolute value) of the complex numbers, or more precisely the Gaussian integers, so the sum of four squares is related to the norm of quaternions, or more precisely the Hurwitz integers.

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