Fermat's Last Theorem fell because two distant branches of mathematics turned out to match
Andrew Wiles did not attack Fermat's Last Theorem head-on. He proved a large part of a claim that elliptic curves, a kind of cubic equation, always correspond to highly symmetric functions called modular forms. Most experts thought that claim hopeless. Once enough of it held, Fermat's centuries-old puzzle collapsed as a consequence.
The statement began as the twelfth of 36 open problems that Yutaka Taniyama posed at a 1955 symposium in Tokyo and Nikkō, in a slightly flawed form. He and Goro Shimura sharpened it until 1957. André Weil rediscovered it and in 1967 supplied the first serious evidence it might be true, also predicting that a curve's conductor should equal the level of its matching modular form. The conjecture, named after all three, became part of Robert Langlands's vast programme linking number theory and symmetry.
What it says is that every elliptic curve over the rational numbers can be reached by a map from a particular modular curve, and that the counts of its solutions modulo each prime line up with the coefficients of a weight-two modular form. For the curve given by y squared plus y equals x cubed minus x, with conductor 37, checking the prime 3 turns up 6 solutions and a matching coefficient of minus 3, just as the theory predicts.
Fermat entered in 1986, when Gerhard Frey suggested that any counterexample to Fermat's claim would produce an elliptic curve too strange to be modular. Jean-Pierre Serre spotted the missing step, and Ken Ribet proved it. Suddenly Fermat was a corollary of the conjecture, making it a respectable research project. Yet Wiles's own doctoral supervisor, John Coates, thought it seemed impossible to prove, and Ribet counted himself among the vast majority who thought it inaccessible. Drawing on his expertise in Iwasawa theory, Wiles, with help from Richard Taylor, proved it in 1995 for semistable curves, which was enough for Fermat.
His former students Brian Conrad and Fred Diamond, with Taylor and Christophe Breuil, then worked through the remaining cases, completing the full result by 1999; it has been called the modularity theorem since. Related results follow, such as the fact that no cube is a sum of two coprime nth powers for n of at least 3. In 2025, Boxer, Calegari, Gee and Pilloni extended modularity to over 10 percent of abelian surfaces.
Source: Modularity theorem
