Is there any natural number $A$ which cannot be written as:
$$A=W^2+X^2+Y^2+Z^2$$
where $W,X,Y,Z \in \mathbb N \cup 0$
I was considering the fact that $a^2+b^2 \not = 1 \mod 4$ and was attempting to determine simular results for $3$ and $4$ squares when I realised that I could not find a number which did not work for $4$. I did a quick search and found that no numbers less than $1000$ contradict this. I have been trying to find a proof but have been, so far, unsuccessful. This is too general a concept to be new but I have not been exposed to it previously.
Best Answer
No, all numbers can be expressed as the sum of four squares. More information, and a proof using the Hurwitz integers, can be found here.