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.