Editing Lagrange's four-square theorem (section)

==Generalizations==
Lagrange's four-square theorem is a special case of the [[Fermat polygonal number theorem]] and [[Waring's problem]]. Another possible generalization is the following problem: Given [[natural number]]s <math>a,b,c,d</math>, can we solve

<math display="block">n=ax_1^2+bx_2^2+cx_3^2+dx_4^2</math>

for all positive integers {{mvar|n}} in integers <math>x_1,x_2,x_3,x_4</math>? The case <math>a=b=c=d=1</math> is answered in the positive by Lagrange's four-square theorem. The general solution was given by [[Ramanujan]].<ref>{{harvnb|Ramanujan|1916}}.</ref> He proved that if we assume, without loss of generality, that <math>a\leq b\leq c\leq d</math> then there are exactly 54 possible choices for <math>a,b,c,d</math> such that the problem is solvable in integers <math>x_1,x_2,x_3,x_4</math> for all {{mvar|n}}. (Ramanujan listed a 55th possibility <math>a=1,b=2,c=5,d=5</math>, but in this case the problem is not solvable if <math>n=15</math>.<ref>{{harvnb|Oh|2000}}.</ref>)