Within the context of discretization of space I stumbled across distances that cannot be represented on a 3D lattice and I wondered if there is a name for them. Let me illustrate it with an example: The distance 1 can be depicted by a 3D lattice as for the combination: (1 0 0) $\sqrt(1^2 +0^2 +0^2)$ and the following distances analogously:
$\sqrt(2)$ >> $\sqrt(1^2 + 1^2 + 0^2)$
$\sqrt(3)$ >> $\sqrt(1^2 + 1^2 + 1^2)$
$\sqrt(4)$ >> $\sqrt(2^2 + 0^2 + 0^2)$
$\sqrt(5)$ >> $\sqrt(2^2 + 1^2 + 0^2)$
$\sqrt(6)$ >> $\sqrt(2^2 + 1^2 + 1^2)$
$\sqrt(7)$ >> $\sqrt(-----)$
$\sqrt(8)$ >> $\sqrt(2^2 + 2^2 + 0^2)$
$\sqrt(9)$ >> $\sqrt(3^2 + 0^2 + 0^2)$
I think the scheme is clear. Is there probably some name for them, for numbers like 7?
These are numbers that cannot be expressed as the sum of $3$ perfect squares.
Legendre's $3$-Square Theorem says that the diophantine equation $n=x^2+y^2+z^2$ has integer solutions for all integer $n$ such that $n$ is NOT of the form $n=4^a(8b+7)$ for integers $a $ and $b$.
The first few numbers that cannot be expressed as the sum of $3$ squares are $7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71...$
Legendre's $4$-Square Theorem guarantees that the diophantine equation $n=x^2+y^2+z^2+w^2$ has integer solutions for all integer $n$.