In 1637, French mathematician Pierre de Fermat proposed, what was to become known as Fermat's Last Theorem: no whole number solutions are possible for Pythagoras' equation for power values greater than two. Unfortunately his notebook's margin wasn't big enough to detail the claimed proof for this. In 1955 Goro Shimura and Yutaka Taniyama met at a mathematics symposium in Tokyo, and proposed The Taniyama-Shimura Conjecture: All elliptical equations have a modular form. And in 1984, at another mathematics symposium, this time in Oberwolfach ( a town in Germany's black-forest), Gerhard Frey - by assuming that there was one whole number solution to Fermat's Enigma - demonstrated how Fermat's Enigma could be expressed as an elliptical equation. One without a corresponding modular form.
Four years later, Andrew Wiles, a Cambridge mathematician working in America, who specialised in matters relating to the Taniyama-Shimura conjecture, and with an interest - from childhood - in Fermat's Enigma, came along. The spur for Wiles was that the work done by Gerhard Frey cast doubt upon the validity of the Taniyama-Shimura Conjecture. This was problematic because Wiles' Ph.D. was in Elliptical equations and Modular forms, a branch of mathematics that other areas of modern work depended upon. And The Taniyama-Shimura Conjecture like Fermat's Enigma had no proof, but was taken to be probably true. So for Wiles, the principal aim was to prove the Taniyama-Shimura Conjecture from which he could also claim to have proved Fermat's Enigma by contradicting Frey's proposition. This he did by 1995, after more than seven years of near solitary toil.
Wiles asserts that Fermat couldn't have had a proof, because the required math didn't exist in 1637: the problem needed twentieth century techniques. This makes sense due to the lack of a proof for over three hundred and fifty years by many others' attempts. Yet why would Fermat have lied? After all, Fermat is regarded as being one of the finest mathematicians there was. Maybe there is another way… someone?
This report prepared by Charles Smyth