It states that every positive integer can be expressed as the sum of at most four squares.

More formally, for every positive integer n there exist non-negative integers a,b,c,d such that *n* = *a*^{2} + *b*^{2} + *c*^{2} + *d*^{2}

Adrien-Marie Legendre improved on the theorem in 1798 by stating that a positive integer can be expressed as the sum of at most three squares if and only if it is not of the form (4^{k})(8*l*-7). His proof was incomplete, leaving a gap which was later filled by Karl Friedrich Gauss.