In —, Gerhard Frey called attention to the unusual properties of this same curve, now called a Frey curve. Mathematically, the conjecture says that each elliptic curve with rational coefficients can be constructed in an entirely different way, not by giving its equation but by using modular functions to parametrise coordinates x and y of the points on it. In , Jean-Pierre Serre provided a partial proof that a Frey curve could not be modular. However his partial proof came close to confirming the link between Fermat and Taniyama. His article was published in

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Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts. The details and auxiliary arguments, however, were often ad hoc and tied to the individual exponent under consideration. Although she developed many techniques for establishing the non-consecutivity condition, she did not succeed in her strategic goal. His proof failed, however, because it assumed incorrectly that such complex numbers can be factored uniquely into primes, similar to integers.

This gap was pointed out immediately by Joseph Liouville , who later read a paper that demonstrated this failure of unique factorisation, written by Ernst Kummer. Kummer set himself the task of determining whether the cyclotomic field could be generalized to include new prime numbers such that unique factorisation was restored.

He succeeded in that task by developing the ideal numbers. Proofs of individual exponents by their nature could never prove the general case: even if all exponents were verified up to an extremely large number X, a higher exponent beyond X might still exist for which the claim was not true. This had been the case with some other past conjectures, and it could not be ruled out in this conjecture.

Main article: Modularity theorem Around , Japanese mathematicians Goro Shimura and Yutaka Taniyama observed a possible link between two apparently completely distinct branches of mathematics, elliptic curves and modular forms. The resulting modularity theorem at the time known as the Taniyama—Shimura conjecture states that every elliptic curve is modular , meaning that it can be associated with a unique modular form. Therefore if the latter were true, the former could not be disproven, and would also have to be true.

This was widely believed inaccessible to proof by contemporary mathematicians. Frey showed that this was plausible but did not go as far as giving a full proof. The modularity theorem — if proved for semi-stable elliptic curves — would mean that all semistable elliptic curves must be modular. Since his work relied extensively on this approach, which was new to mathematics and to Wiles, in January he asked his Princeton colleague, Nick Katz , to help him check his reasoning for subtle errors.

Their conclusion at the time was that the techniques Wiles used seemed to work correctly. However, it became apparent during peer review that a critical point in the proof was incorrect. It contained an error in a bound on the order of a particular group. Wiles spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former student Richard Taylor , without success. Mathematicians were beginning to pressure Wiles to disclose his work whether it was complete or not, so that the wider community could explore and use whatever he had managed to accomplish.

But instead of being fixed, the problem, which had originally seemed minor, now seemed very significant, far more serious, and less easy to resolve.

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## Fermat's Last Theorem

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## Wiles's proof of Fermat's Last Theorem

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