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The conjecture attracted considerable interest when [[Gerhard Frey (mathematician)|Gerhard Frey]]{{sfn|Frey|1986}}<!--{{harvs|txt|authorlink=Gerhard Frey (mathematician)|last=Frey|first=Gerhard|year=1986}}--> suggested in 1986 that it implies [[Fermat's Last Theorem]]. He did this by attempting to show that any counterexample to Fermat's Last Theorem would imply the existence of at least one non-modular elliptic curve. This argument was completed in 1987 when Jean-Pierre Serre{{sfn|Serre|1987}}<!--{{harvs|txt|authorlink=Jean-Pierre Serre|last=Serre|first=Jean-Pierre|year=1987}}--> identified a missing link (now known as the [[epsilon conjecture]] or Ribet's theorem) in Frey's original work, followed two years later by Ken Ribet's completion of a proof of the epsilon conjecture.{{sfn|Ribet|1990}}<!--{{harvs|txt|authorlink=Ken Ribet|last=Ribet|first=Ken|year=1990}}-->
Even after gaining serious attention, the Taniyama–Shimura–Weil conjecture was seen by contemporary mathematicians as extraordinarily difficult to prove or perhaps even inaccessible to proof<!--{{harv|Singh|1997|pp=203–205, 223, 226}}-->.{{sfn|Singh|1997|pp=203–205, 223, 226}} For example, Wiles's Ph.D. supervisor [[John H. Coates|John Coates]] states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".
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:<math>\Delta = \frac{1}{256}(abc)^{2p}</math>
cannot be modular.{{sfn|Ribet|1990}} Thus, the proof of the Taniyama–Shimura–Weil conjecture for this family of elliptic curves (called Hellegouarch–Frey curves) implies FLT. The proof of the link between these two statements, based on an idea of [[Gerhard Frey (mathematician)|Gerhard Frey]] (1985), is difficult and technical. It was established by [[Kenneth Ribet]] in 1987.<ref>See the survey of {{cite journal |first=K. |last=Ribet |title=From the Taniyama–Shimura conjecture to Fermat's Last Theorem |journal=Annales de la Faculté des Sciences de Toulouse |volume=11 |year=1990b |pages=116–139 |doi= 10.5802/afst.698|url=http://www.numdam.org/item?id=AFST_1990_5_11_1_116_0 |doi-access=free }}</ref>
==Notes==
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