This would finally prove Fermat's Last Theorem. The contradiction could have only one answer: if Ribet's Theorem and the Taniyama–Shimura–Weil conjecture for semistable curves were both true, then it would mean there could not be any solutions to Fermat's equation – because then there would be no Frey curves at all, meaning no contradictions would exist.But if the Taniyama–Shimura–Weil conjecture were also true for semistable elliptic curves, then by definition every Frey's curve that existed must be modular.From Ribet's Theorem and the Frey Curve, any 4 numbers able to be used to disprove Fermat's Last Theorem could also be used to make a semistable elliptic curve ('Frey's curve') that could never be modular.Precursors to Wiles' proof Fermat's Last Theorem and progress prior to 1980 įollowing the developments related to the Frey Curve, and its link to both Fermat and Taniyama, a proof of Fermat's Last Theorem would follow from a proof of the Taniyama–Shimura–Weil conjecture - or at least a proof of the conjecture for the kinds of elliptic curves that included Frey's equation (known as semistable elliptic curves). 3 Announcement and subsequent developments.When announcing that Wiles had won the Abel Prize, the Norwegian Academy of Science and Letters described his achievement as a 'stunning proof'. For solving Fermat's Last Theorem, he was knighted, and received other honours such as the 2016 Abel Prize. Wiles' path to proving Fermat's Last Theorem, by way of proving the modularity theorem for the special case of semistable elliptic curves, established powerful modularity lifting techniques and opened up entire new approaches to numerous other problems. John Coates described the proof as one of the highest achievements of number theory, and John Conway called it the proof of the century. Together, the two papers which contain the proof are 129 pages long, and consumed over seven years of Wiles's research time.

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