Equations of the form x^n + y^n = z^n, called Diophantine equations when n is an integer, can be translated to describe a certain set of elliptic curves. These curves represent the surface of a torus, an object shaped like a smooth doughnut.

Taniyama suggested that for that certain set of elliptic curves in Euclidean geometry (where parallel lines never meet, even if infinitely extended) there are corresponding structures in the hyperbolic (non-Euclidian) plane (where parallel lines can both converge and diverge).

Frey suggested a connection between that certain set of elliptic curves and Fermat's last theorem, namely that if there were solutions in violation of the theorem, they would generate a subset of **"semistable" elliptic curves, curves that could not be represented in the hyperbolic plane**.

Wiles accepted Ribet's proof of Frey and reasoned that if he (Wiles) could prove Taniyama, at least for the "Fermat subset" of semistable elliptic curves (if not for that larger certain set of elliptic curves), no solutions to Fermat's last theorem could exist, thereby implying a proof of FLT.

Wiles then developed an unconventional method of counting both the Euclidian semistable elliptic curves and their hyperbolic (non-Euclidian) counterparts in such a way as to demonstrate a one-to-one correspondence between the two groups. In this way, he claims to have proved Taniyama for the "Fermat subset" of semistable elliptic curves.