For which positive integers $$a, b, c, d$$, any natural number $$n$$ can be represented as

$n=ax^2+by^2+cz^2+dw^2,$

where $$x, y,z,w$$ are integers?

Lagrange’s four-square theorem states that $$(a,b,c,d)=(1,1,1,1)$$ works. Ramanujan proved that there are exactly $$54$$ possible choices for $$a, b, c, d$$.

For which positive integers $$a, b, c, d$$,

$n=ax^2+by^2+cz^2+dw^2,$

is solvable in integers $$x, y,z,w$$ for all positive integers $$n$$ except one number? For example, $$n=x^2+y^2+2z^2+29w^2$$ is solvable for all natural number $$n$$ except $$14$$, $$n=x^2+2y^2+7z^2+11w^2$$ and $$n=x^2+2y^2+7z^2+13w^2$$ except $$5$$.

P.R.Halmos proved that there are exactly $$88$$ possible choices for $$a, b, c, d$$.

What integers are not in the range of $$a^2+b^2+c^2-x^2$$? Ramanujan also thought about that.