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In 1985–86, Wiles was a Guggenheim Fellow at the Institut des Hautes Études Scientifiques near Paris and at the École Normale Supérieure.

From 1988 to 1990, Wiles was a Royal Society Research Professor at the University of Oxford, and then he returned to Princeton.

He threatened me by stating that if I showed up in Court he would make sure the Prosecutor would have additional charges waiting for me.

Together these colleagues worked on the arithmetic of elliptic curves with complex multiplication by the methods of Iwasawa theory.

However, he soon realised that his knowledge was too limited, so he abandoned his childhood dream, until it was brought back to his attention at the age of 33 by Ken Ribet's 1986 proof of the epsilon conjecture, which Gerhard Frey had previously linked to Fermat's famous equation.

After a stay at the Institute for Advanced Study in Princeton, New Jersey, in 1981, Wiles became a Professor of Mathematics at Princeton University.

Starting in mid-1986, based on successive progress of the previous few years of Gerhard Frey, Jean-Pierre Serre and Ken Ribet, it became clear that Fermat's Last Theorem could be proven as a corollary of a limited form of the modularity theorem (unproven at the time and then known as the "Taniyama–Shimura–Weil conjecture").

The modularity theorem involved elliptic curves, which was also Wiles's own specialist area. In August 1993, it was discovered that the proof contained a flaw in one area.

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and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible", adding that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]." He dedicated all of his research time to this problem for over six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife. Wiles concluded that he had proved a general case of the Taniyama conjecture. Wiles tried and failed for over a year to repair his proof.