Legendre's three-square theorem
In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers if and only if is not of the form formula_2 for nonnegative integers and . The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as formula_2) are History. Pierre de Fermat gave a criterion for numbers of the form 8"a" + 1 and 8"a" + 3 to be sums of a square plus twice another square, but did not provide a proof . N. Beguelin noticed in 1774 that every positive integer which is neither of the form 8"n" + 7, nor of the form 4"n", is the sum of three squares, but did not provide a satisfactory proof. In 1796 Gauss proved his Eureka theorem that every positive integer "n" is the sum of 3 triangular numbers; this is equivalent to the fact that 8"n" + 3 is a sum of three squares. In 1797 or 1798 A.-M. Legendre obtained the first proof of his 3 square theorem. In 1813, A. L. Cauchy noted that Legendre's theorem is equivalent to the  statement in the introduction above. Previously, in 1801, C. F. Gauss had obtained a more general result, containing Legendre's theorem of 1797–8 as a corollary. In particular, Gauss counted the number of solutions of the expression of an integer as a sum of three squares, and this is a generalisation of yet another result of Legendre, whose proof is incomplete. This last fact appears to be the reason for later incorrect claims according to which Legendre's proof of the three-square theorem was defective and had to be completed by Gauss. With Lagrange's four-square theorem and  the two-square theorem of Girard, Fermat and Euler, the Waring's problem for "k" = 2 is entirely solved. Proofs. The "only if" of the theorem is simply because modulo 8, every square is congruent to 0, 1 or 4. There are several proofs of the converse (besides Legendre's proof). One of them is due to J. P. G. L. Dirichlet in 1850, and has become classical. It requires three main lemmas: Relationship to the four-square theorem. This theorem can be used to prove Lagrange's four-square theorem, which states that all natural numbers can be  written as a sum of four squares. Gauss pointed out that the four squares theorem follows easily from the fact that any positive integer that is 1 or 2 mod 4 is a sum of 3 squares, because any positive integer not divisible by 4 can be reduced to this form by subtracting 0 or 1 from it. However, proving the three-square theorem is considerably more difficult than a direct proof of the four-square theorem that does not use the three-square theorem. Indeed, the four-square theorem was proved earlier, in 1770. Legendre's three-square theorem