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.^[1] That is, the squares form an additive basis of order four: $${\displaystyle p=a^{2}+b^{2}+c^{2}+d^{2},}$$ where the four numbers ${\displaystyle a,b,c,d}$ are integers. For illustration, 3, 31, and 310 can be represented as the sum of four squares as follows: $${\displaystyle {\begin{aligned}3&=1^{2}+1^{2}+1^{2}+0^{2}\\[3pt]31&=5^{2}+2^{2}+1^{2}+1^{2}\\[3pt]310&=17^{2}+4^{2}+2^{2}+1^{2}\\[3pt]&=16^{2}+7^{2}+2^{2}+1^{2}\\[3pt]&=15^{2}+9^{2}+2^{2}+0^{2}\\[3pt]&=12^{2}+11^{2}+6^{2}+3^{2}.\end{aligned}}}$$

This theorem was proven by Joseph Louis Lagrange in 1770. It is a special case of the Fermat polygonal number theorem.