An identity which applies to all of the natural numbers Prove that any natural number n can be written as $$n=a^2+b^2-c^2$$ where $a,b,c$ are also natural.

Consider $n\ge 6$.
If $n$ is odd, $n=2m+1$, then
$$
n = 2m+1 = 2^2 + (m-1)^2 - (m-2)^2;
$$
If $n$ is even, $n=2m$, then
$$
n = 2m = 1^2 + m^2 - (m-1)^2;
$$
Small $n$:
$1=1^2+1^2-1^2$,
$2=3^2+3^2-4^2$,
$3=4^2+6^2-7^2$,
$4=2^2+1^2-1^2$,
$5=4^2+5^2-6^2$.
