Lagrange’s Four-square Theorem

Lagrange’s four-square theorem, also called Lagrange’s theorem, in number theory, theorem that every positive integer can be expressed as the sum of the squares of four integers. For example,23 = 1² + 2² + 3² + 3². The four-square theorem was first proposed by the Greek mathematician Diophantus of Alexandria in his treatise Arithmetica (3rd century ce). Credit for the first proof is given to the 17th-century French amateur mathematician Pierre de Fermat. (Although he did not publish this proof, his study of Diophantus led to Fermat’s last theorem.) The first published proof of the four-square theorem was in 1770 by the French mathematician Joseph-Louis Lagrange, for whom the theorem is now named.

The impetus for renewed interest in Diophantus and such problems in number theory was the Frenchman Claude-Gaspar Bachet de Méziriac, whose Latin translation Diophanti (1621) of Arithmetica brought the work to a wider audience. In addition to the proof of Diophantus’s four-square theorem, study of the text led to a generalization of the theorem known as Waring’s problem.