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The question. Can every $n\in \mathbb N$ can be written:
$$n=a^2\pm b^2\pm c^2$$
where $\pm$ are signs of your choice?
We know with Lagrange's four-square theorem that every integer can be written as the sum of four squares.
Plus, with have Legendre's three-square theorem stated that an integer can not be written as the sum of three squares if, and only if, it is of the form:
So we just have to prove (or disprove) it for every number of this form.
I have checked it until $55$, and it seems to work so far. So the number we have to check are these ones.
The issue here is that $a$, $b$ and $c$ can be arbitrarily large. For instance:
So I don't really know how to prove or disprove this result, and I think it could go either way.