The number 1729 is part of the Ramanujan lore, being famously the number of the taxi which Hardy took to visit Ramanujan. In this talk I would like to argue that the prime number 691 is a number of greater significance than 1729 to the mathematics that Ramanujan discovered. It occurs in his paper On certain arithmetical functions published in 1916 in which he observed a congruence between modular forms modulo 691. The observations Ramanujan made in that paper had a great influence on the developments in number theory in the 20th century which led to a proof of Fermat’s Last Theorem by Andrew Wiles. The 1916 observations of Ramanujan led to Serre formulating his influential modularity conjecture which was proved in 2009 by Jean-Pierre Wintenberger and myself. I will explain some of these developments and current work on the connection between modular forms and Galois representations.
This video was produced by the Chennai Mathematical Institute as part of the workshop Perspectives in Mathematical Sciences.
