Lagrange's 4-square theorem

number-theory

what

Every natural number can be represented as a sum of 4 non-negative integer squares

$$n=a^2+b^2+c^2+d^2$$

where $n>0$ and $a,b,c,d$ are integers.

a.k.a  Bachet’s conjecture

properties

• a special case of the Fermat polygonal number theorem.
• according to Legendre’s 3-square theorem, all 4 squares must be positive if $n=4^k(8m+7)$ for integers $k$ and $m$.