Editing Lagrange's four-square theorem (section)

{{short description|Every natural number can be represented as the sum of four integer squares}}
{{for multi|Lagrange's identity|Lagrange's identity (disambiguation)|Lagrange's theorem|Lagrange's theorem (disambiguation)}}
{{Redirect2|four-square theorem|four square theorem|other uses|four square (disambiguation)}}
[[File:distances_between_double_cube_corners.svg|thumb|Unlike in three dimensions in which distances between [[Vertex (geometry)|vertices]] of a [[polycube]] with unit edges excludes √7 due to [[Legendre's three-square theorem]], Lagrange's four-square theorem states that the analogue in four dimensions yields [[square root]]s of every [[natural number]] ]]

'''Lagrange's four-square theorem''', also known as '''Bachet's conjecture''', states that every [[natural number|nonnegative integer]] can be represented as a sum of four non-negative integer [[square number|squares]].{{r|andrews}} That is, the squares form an [[additive basis]] of order four:
<math display="block">p = a^2 + b^2 + c^2 + d^2,</math> 
where the four numbers <math>a, b, c, d</math> are integers. For illustration, 3, 31, and 310 can be represented as the sum of four squares as follows:
<math display="block">\begin{align}
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{align}</math>

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