Legendre's 3-square theorem

number-theory

what

A natural number can be represented as the sum of the squares of 3 integers $n=a^2+b^2+c^2$ if and only if $n\ne 4^k(8m+7)$ for non-negative integers $k$ and $m$.

properties

• can be used to proof Lagrange’s 4-square theorem, but it’s harder to proof that a direct proof of Lagrange’s 4-square theorem.