: Mathématiques: Lagrange's four-square theorem

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Lagrange's four-square theorem

Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as a sum of four non-negative integer squares. 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 in several ways, can be represented as the sum of four squares as follows: ${\\displaystyle {\egin{aligned}3&=1^{2}+1^{2}+1^{2}+0^{2}\\\\31&=5^{2}+2^{2}+1^{2}+1^{2}\\\\310&=17^{2}+4^{2}+2^{2}+1^{2}\\\\&=16^{2}+7^{2}+2^{2}+1^{2}\\\\&=15^{2}+9^{2}+2^{2}+0^{2}\\\\&=12^{2}+11^{2}+6^{2}+3^{2}.\\end{aligned}}}$ This theorem was proven by Joseph Louis Lagrange in 1770. It is a special case of the Fermat polygonal number theorem.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem)

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