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Fermat’s Last Conjecture

Fermat’s Last Conjecture

By Aditya Chopra

The Simpsons is a show notoriously known to throw in mathematical Easter eggs in their scenes. Since their off screen cast has an astonishingly high number of mathematicians and mathematics aficionados the show often features such throwbacks to their mathematical nature. One recurring Easter egg is their attempt to disprove Fermat’s Last Theorem.

The Theorem simply states that for any natural numbers a, b and c

an +bn≠cn ; For any n> 2

This effectively means that the above equation holds true for only n=1 or 2.

The above equation is also called Fermat’s last theorem, because it was noticed first by Pierre De Fermat for the first time back in 1637. The French lawyer Fermat on a late night noticed this peculiarity in mathematics and in the margin of the book he was reading wrote:

Hanc marginis exiguitas non caperet

This translates to: I have a truly marvelous proof, which this margin is too narrow to contain. Unfortunately, before he could find a less narrow margin, Fermat passed away leaving the world with an equation no one could prove for nearly 350 years.

Before his death Fermat did prove his theorem for n=4. And in the next three and a half centuries it was proved for n =3, 5, and 7. But no one could find a general proof for the equation. Fermat’s Last Theorem had become Fermat’s Conjecture i.e a question no one could answer. But could Fermat? Fermat Might have been a mathematics fan, but there was no proof that he was bright enought to derive a general proof for this equation.

Nearly 350 years later, the general proof is still not found despite countless attempts by many accomplished mathematicians. A young boy stumbles across the conjecture at an age when the equation would have made little or no sense to him. Andrew Wiles read about Fermat and his mystery and instantly realized that it would be his lifelong objective to crack it. He spoke about it to his high school teachers, peers and soon college professors. After achieving success in his life, Wiles at a much older age came across a special case of Fermat’s Conjecture: the Taniyama- Shimura Conjecture. It just so happened that if he could prove the Taniyama- Shimura Conjecture he would consequently also prove Fermat’s Conjecture.

And so in six years of complete secrecy Andrew Wiles came up with an extensive and vast solution, which helped prove both conjectures. To tell the world about his work he held three lectures at the Isaac Newton Institute for Mathematical Sciences. However in the review of the lecture many mathematicians noticed and informed Wiles of a grave error.

Wiles was determined that he overcome the error and still solve the equation. But in the next year and a half he worked on it Wiles’ error turned out be a much more grave situation and any correction seemed to render his work worthless. The mathematics community was coercing him to release his unfinished work so that more people could have access to it and build on it. On one of the last few days before he did so he sat down for an ultimate review. His resignation was noticeable in his conduct. He had failed his work. And then in almost a movie like moment the solution occurred to him. He realized that his current solution was dependent on another path he had initially rejected. In his own words he described his realization as:

“It was so indescribably beautiful; it was so simple and so elegant.

In a true Eureka moment he had solved a proof, which eluded centuries of mathematicians. Cue the angelic symphonies and laurel wreaths, a 358-year-old monster had been slain.

Rohil Bahl