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In 1995, Andrew Wiles, with some help from [[Richard Taylor (mathematician)|Richard Taylor]], proved the Taniyama–Shimura–Weil conjecture for all [[semistable elliptic curve]]s. Wiles used this to prove Fermat's Last Theorem,{{sfnm|Wiles|1995a|Wiles|1995b}}<!--{{harvs|txt|authorlink=Andrew Wiles|last=Wiles|year=1995}}--> and the full Taniyama–Shimura–Weil conjecture was finally proved by Diamond,{{sfn|Diamond|1996}}<!--{{harvtxt|Diamond|1996}}--> Conrad, Diamond & Taylor; and Breuil, Conrad, Diamond & Taylor; building on Wiles's work, they incrementally chipped away at the remaining cases until the full result was proved in 1999.{{sfn|Conrad|Diamond|Taylor|1999}}<!--{{harvtxt|Conrad|Diamond|Taylor|1999}}-->{{sfn|Breuil|Conrad|Diamond|Taylor|2001}}<!--{{harvtxt|Breuil|Conrad|Diamond|Taylor|2001}}--> Once fully proven, the conjecture became known as the modularity theorem.
Several theorems in number theory similar to Fermat's Last Theorem follow from the modularity theorem. For example: no cube can be written as a sum of two [[coprime]] <math>n</math>-th powers, <math>n \geq 3</math>.
==Generalizations==
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