10/15/2014

Fermat's Last Theorem

The Fermat's Last Theorem states that for $a, b, c \in \mathbb{Z^+}$ and $n \in \mathbb{N}, n > 2$, the equation $ a^n + b^n = c^n$ doesn't hold. 

Despite its apparent simplicity in terms of hypothesis and conclusion, this theorem is actually very involved and remained a mere conjecture until Andrew Wiles formally proved it in 1995, more than 300 years after Fermat first wrote it in the margin of Diophantus's \textit{Arithmetica}. 

The page of \textit{Arithmetica} that motivated the question was about Pythagorean theorem, which states that in any right triangle, the square of hypotenuse is equal to the sum of square of two other sides. In other words, $\exists a,b,c \in \mathbb{Z^+}$ such that $a^2+b^2=c^2$.   

Having tested various solutions to the Pythagorean theorem, Fermat went a step further to check whether the theorem holds for $n=3$. Surprisingly, Fermat argued that such system is insoluble and more generally, there is no positive integer solutions to any n that is larger than $2$. He commented that he came up with a remarkable proof to solve for all cases but the margin is too narrow for him to write down.\cite{citationkey1} As a result, we are only able to read Fermat's proof for a special case where $n=4$, while the general theorem remained unsolved for the next three centuries. 

Mathematicians at first tried to find existence of counter-examples but failed, which incentivized more people to believe that Fermat's conjecture is true. Later, mathematicians managed to prove the conjecture when $n=3, 5$ and $7$, but no one was able to come up with a general proof for all equations. 

The longer one conjecture remained mysterious, the more attractive it seemed to people who inspired to solve it. One story says that in nineteenth century a German physician and mathematician Paul Wolfskehl bequeathed 100,000 marks (quite a big amount of money at that time) to the first one who could prove it. Such lucrative prize motivated more people, especially amateurs to hand in their proofs, but unfortunately none of their attempt was successful. Due to advent of modern computers after WWII, mathematicians were able to conclude that Fermat's conjecture holds for finitely large $n$, but still, nobody was able to give a complete proof. The conjecture also became notorious for its difficulty and before Wiles' proof, it was described as "most difficult mathematical problems" in the \textit{Guinness Book of World Records}. \cite{2}

In the twentieth century, Shimura and Taniyama, two Japanese mathematicians specialized in elliptic curves, conjectured that each elliptic curve can be matched to a modular. It seemed at first Shimura-Taniyama conjecture was unrelated to Fermat's conjecture, but surprisingly, Kennith Ribet and Gerhard Frey reasoned that Shimura-Taniyama conjecture doesn't hold unless there exists no positive integer solutions to $a^n+b^n=c^n$ for infinitely large $n$. In other words, Shimura-Taniyama conjecture begs the question that Fermat's last theorem is true. Hence, the Fermat's conjecture returned to mainstream of math proof not just for intellectual interest, but also for paving way for new branches of math research.   
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Andrew Wiles, who has been obsessed with Fermat's Last Theorem since childhood, specialized in elliptic curves both as a graduate student in Cambridge and as a professor in Princeton. On hearing Ribet and Frey's research progress, he decided to prove Fermat's conjecture by attacking Shimura-Taniyama conjecture. He spent seven years working alone, learning and extending some new methods at that time,\cite{3} and presented his proof in 1993. While Wiles enjoyed public accolade and fame at first, a team of referees found in his paper a big flaw in a bound of the order of a specific group. \cite{4} Wiles spent one more year trying to repair his proof. Fortunately, before he wanted to give up an intuition struck him and help him correct the previous approach. Wiles published his manuscripts in 1995 on \textit{Annals of Mathematics}, and he proved Fermat's Last Theorem by techniques seemingly unrelated to number theory. After over 300 years of effort, mathematicians finally were able to prove Fermat's Last Theorem, and the process of proof is roundabout and interdisciplinary.