Lagrange's four-square theorem

Lagrange's four-square theorem, also known as Bachet's conjecture, states that every nonnegative integer can be represented as a sum of four non-negative integer squares. That is, the squares form an additive basis of order four: p = a 2 + b 2 + c 2 + d 2 , {\displaystyle p=a^{2}+b^{2}+c^{2}+d^{2},} where the four numbers a , b , c , d {\displaystyle a,b,c,d} are integers. For illustration, 3, 31, and 310 can be represented as the sum of four squares as follows: 3 = 1 2 + 1 2 + 1 2 + 0 2 31 = 5 2 + 2 2 + 1 2 + 1 2 310 = 17 2 + 4 2 + 2 2 + 1 2 = 16 2 + 7 2 + 2 2 + 1 2 = 15 2 + 9 2 + 2 2 + 0 2 = 12 2 + 11 2 + 6 2 + 3 2 . {\displaystyle {\begin{aligned}3&=1^{2}+1^{2}+1^{2}+0^{2}\\[3pt]31&=5^{2}+2^{2}+1^{2}+1^{2}\\[3pt]310&=17^{2}+4^{2}+2^{2}+1^{2}\\[3pt]&=16^{2}+7^{2}+2^{2}+1^{2}\\[3pt]&=15^{2}+9^{2}+2^{2}+0^{2}\\[3pt]&=12^{2}+11^{2}+6^{2}+3^{2}.\end{aligned}}} This theorem was proven by Joseph-Louis Lagrange in 1770. It is a special case of the Fermat polygonal number theorem.