A history of Lagrange's four-square theorem and related developments

Time: 6:00pm

Location: 210 St Augustine Ctr, Villanova University

In 1770 Lagrange proved his now famous theorem that every positive integer can be written as sum of four squares of integers. Fermat had already observed this result and Euler made enormous efforts to prove this theorem. Neither succeeded in finding a proof. Within two years of Lagrange proving this theorem, Euler gave a simplification of Lagrange's proof. It is Euler's proof that has become the standard textbook proof. Lagrange's proof is now largely forgotten. A careful analysis shows that Lagrange's proof can be used to prove a much stronger theorem, foreshadowing a theorem proved a hundred years later in the early 1870's. In this talk I plan to give a history of the results and partial results found by Fermat and Euler that ultimately led Lagrange to his proof. I will also mention the extension of Lagrange's theorem mentioned above. My intent is to make this talk accessible to people with a wide variety of mathematical backgrounds.