T4.JPGIn 1963, when he was a ten-year-old, Andrew Wiles found a copy of a book on Fermat’s Last Theorem in his local library. He became captivated by it which was easy to understand but which had remained unsolved for 300 years. “I knew from that moment that I would never let it go,” he said. “I had to solve it.”  Fermat’s Last Theorem, first formulated by Pierre de Fermat in the 17th century, is the assertion that the equation x^n + y^n = z^n has no solutions in positive integers for n>2.  Fermat proved his claim for n=4, Leonhard Euler found a proof for n=3, and Sophie Germain proved the first general result that applies to infinitely many prime exponents.  The complete proof found by Andrew Wiles relies on three further concepts in number theory, namely elliptic curves, modular forms, and Galois representations.  Elliptic curves are defined by cubic equations in two variables. They are the natural domains of definition of the elliptic functions introduced by Niels Henrik Abel.  Modular forms are highly symmetric analytic functions defined on the upper half of the complex plane, and naturally factor through shapes known as modular curves. An elliptic curve is said to be modular if it can be parameterized by a map from one of these modular curves.  Andrew Wiles, in a breakthrough paper published in 1995, introduced his modularity lifting technique and proved the semistable case of the modularity conjecture and also Fermat’s Last Theorem..  Few results have as rich a mathematical history and as dramatic a proof as Fermat’s Last Theorem.  Wiles’ proof was an epochal moment for mathematics and also the culmination of a remarkable personal journey that began three decades before.