Fermat’s Last Theorem: The compelling biography and history of mathematical intellectual endeavour

by Simon Singh

Therefore the existence of Frey’s elliptic equation defies the Taniyama–Shimura conjecture. In other words, Frey’s argument was as follows: (1) If (and only if) Fermat’s Last Theorem is wrong, then Frey’s elliptic equation exists. (2) Frey’s elliptic equation is so weird that it can never be modular. (3) The Taniyama–Shimura conjecture claims that every elliptic equation must be modular. (4) Therefore the Taniyama–Shimura conjecture must be false! Alternatively, and more importantly, Frey could run his argument backwards: (1) If the Taniyama–Shimura conjecture can be proved to be true, then every elliptic equation must be modular. (2) If every elliptic equation must be modular, then the Frey elliptic equation is forbidden to exist. (3) If the Frey elliptic equation does not exist, then there can be no solutions to Fermat’s equation. (4) Therefore Fermat’s Last Theorem is true!