Lagrange's proof of Bachet's Conjecture

Background. The theorem originated in a 1621 Latin translation by Claude Gaspard Bachet de Méziriac, who stated the conjecture in his notes. Although the statement was known, it lacked a formal demonstration. Joseph-Louis Lagrange, an Italian and naturalized French mathematician, had succeeded Euler as director of mathematics at the Prussian Academy of Sciences in Berlin in 1766, where he stayed for over twenty years.

The Proof. Lagrange proved in 1770 that every positive integer is the sum of four squares. This result is also described as proving that every nonnegative integer can be represented as a sum of four non-negative integer squares. The proof established that the squares form an additive basis of order four, confirming the conjecture that had remained unproved for over a century.

Mathematical Context. This achievement was part of Lagrange's broader contributions to number theory, including his discussion of representations of integers by quadratic forms in 1769 and algebraic forms in 1770. The four-square theorem is a special case of the Fermat polygonal number theorem and Waring's problem. Later, in 1834, Carl Gustav Jakob Jacobi discovered a formula for the number of such representations.