Fermat's Last Theorem

Sometimes the simplest of circumstances can lead to monumental stumbling blocks. Ask anyone who works in the repair business; they'll have at least one story of how a five-minute problem ballooned into three hours of work. The same goes for delivery drivers, doctors, plumbers...and mathematicians. A prime example of this would be Pierre de Fermat, a 17^th century number theorist who, perhaps inadvertently, created a mathematical conundrum that stood as one of the great unsolved problems of the field for 350 years.

Fermat, who was actually a lawyer (his mathematical pursuits were apparently something he did on the side), had some interesting theories on how best to pursue his research. During his lifetime, he only published one paper, and that was done anonymously. It was in the interest of keeping Fermat's research from vanishing into obscurity that prompted his son Samuel to compile all of his letters and notes, and here is where the fun begins. In his father's copy of Arithmetica by Diophantes, Samuel found a handwritten note in one of the book's margins, stating that the equation

xⁿ + yⁿ = zⁿ

does not have a non-zero integer solution for x, y, and z when n is greater than 2. This note, which was jotted down somewhere between 1630 and 1637, became known as Fermat's Last Theorem. To further add mystery, Fermat further noted below the Theorem's statement that he had "discovered a truly remarkable proof which this margin is too small to contain." Unfortunately, none of his other notes or papers contained this proof, or even a hint of it. When Fermat died in 1665, there was still no trace of this "remarkable proof."

Over the next 350 years, many mathematicians had a go at cracking this deceptively simple-seeming problem. While special cases for n=3 and n=4 were discovered, nobody had yet made a proof for the general case which stood up, despite the sheer number of stellar mathematical minds which had focused on the problem, including Leonhard Euler, Sophie Germain and Lejeune Dirichlet. Despite the fame that attached to the problem (and the prize money that was offered by various agencies to successfully solve it), it stood unresolved. (Note: for an excellent breakdown of the history of Fermat's Last Theorem, visit http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html.)

Then, in 1986, it was announced that a connection between an earlier conjecture known as the Taniyama-Shimura-Weil conjecture and Fermat's theorem had been proven, which piqued the interest of British mathematician Andrew Wiles. Wiles, who had been interested in FLT since childhood, realized that by proving the Taniyama-Shimura conjecture, which concerned elliptical curves and the properties of the space which they inhabited, he could also construct a proof of FLT. For the next seven years, Wiles worked on FLT in secret, fearing any news of his efforts would draw attention and distract him from the problem at hand.

Finally, while working at Princeton in 1993, Wiles was able to complete a proof of the Taniyama-Shimura conjecture for an entire class of curves, which covered those mentioned in FLT. Although the FLT proof was actually a corollary of his main work, it quickly garnered more attention. Announced in June 1993, it made headlines in the press and waves in the mathematical community, particularly in October 1993, when Wiles announced that he had found a flaw in his work. For the next year, he worked in collaboration with another mathematician, Richard Taylor, to repair the flaw and validate the original argument, and in October 1994, was able to do so. (An interview with Andrew Wiles done for NOVA can be found at http://www.pbs.org/wgbh/nova/proof/wiles.html.)

Notably, the proof was approximately 150 pages long, and of such complexity that it took many months to fully validate. Due to the complexity of the proof and the aspects of different mathematical fields that it touches, it is commonly believed that Fermat did not have a working proof for his theorem, but only discovered this after making his famous notation. Unlike his last theorem, this is a conjecture that will most likely never be proven one way or the other.