Lagrange's four-square theorem

Statement

Each natural number $n$ is a sum of four squares of integers: $n = a^2 + b^2 + c^2 + d^2$ for some integers $a$, $b$, $c$ and $d$.

Remarks

For example, $7 = 4 + 1 + 1 + 1$ and $15 = 9 + 4 + 1 + 1$. Three squares are not enough: $7$ is not a sum of three squares, by Legendre's three-square theorem. By Euler's four-square identity, it is enough to prove the theorem for primes. Other proofs use Minkowski's theorem or the Hurwitz quaternions.

History

Diophantus knew examples, and Bachet stated the theorem in his Latin translation of the Arithmetica of 1621. Lagrange gave the first proof in 1770.

Explanations

L1 human

Every whole number is a sum of at most four perfect squares, if you allow $0$ as a square. Some numbers, such as $7$, $15$ and $23$, need all four.