# The enigma of Fermat's last theorem

*A member of the IAS, Jaspalkaler loves to read and write on current issues. He also loves mathematics and economics particularly game theory*

## A fascinating and intriguing problem

For 350 years mathematicians were both intrigued and fascinated by Fermat's last theorem, FLT for short. It says a Pythagorean of order higher than two admits of no non-zero integral solutions.

By the way Pythagoras theorem says square of the hypoteneuse of a right triangle is equal to the sum of squares of the other two sides. Symbolically x^{2 }+ y^{2 }= z^{2}. This equation is a Pythagorean of order two. On the same analogy a Pythagorean of order n is an equation wherein sum of nth powers of two numbers is equal to another number raised to the power n i.e. x* ^{n }*+ y

*= z*

^{n }*.*

^{n}A Pythagorean of second order admits integral solutions. So we can find positive integers which satisfy it. The simplest solution of a Pythagorean of second order consists of 3, 4 and 5. The beauty of this solution is that these are three consecutive integers.

## Fermat postulates

Pierre de Fermat, a jurist by profession, who lived in seventeenth century France, postulated that a Pythagorean of order higher than two admits of no non-zero integral solutions. He claimed he had a truly remarkable demonstration of the proof of the theorem which according to him was too lengthy for the margins of the book *Arithmatica* by *Diophantus* that he was reading. He published a paper only once in his life-time and that too anonymously as an appendix in a colleagues' work as he was morally opposed to using his research in any ways commercial. His proofs and methods therefore were not always known. And thus began the saga of a long search for the proof of Fermat's last theorem.

## The hardest mathematical problem ever

The beauty of the theorem lies in its simplicity, but the simplest postulates are often not the easiest to prove or even to accept. For 350 years mathematicians tried to prove or disprove his theorem without success. Euler to Gauss, Legendre to Liouville to Gallois everyone tried it without success. Mathematics history is replete with a chequered list of anecdotes on attempts made to prove this simple looking statement.

## Andrew Wiles finally cracks it

Andrew Wiles, an English mathematician finally did it in 1995. But he used advanced analytical methods using properties of elliptical curves and proved Shimura-Taniyama conjecture and deduced FLT. Such analytical methods were not available to Fermat. So the mystery of Fermat's proof remains and continues to baffle mathematicians. If Fermat had a proof it would be pretty simple and straight forward. And the search for Fermat's proof continues even today.

Gauss, himself one of the greatest of mathematicians and a number theorist, is said to have shown some contempt for the problem. He regarded FLT as an isolated conjecture which can neither be proved nor disproved and therefore he would rather engage himself in more structured and serious mathematics. It was left to Jean-Pierre Serre and Ribet to show a link between FLT and the Shimura-Taniyama conjecture establishing a relationship between elliptic curves and FLT. This demonstrated that Fermat's postulate was none of the non-serious mathematics and has great practical value.

## Invisible lady of mathematics

Story of Marie Sophie Germain, a french mathematician of eighteenth century, in her quest for FLT proof merits particular mention. As women in that era were not encouraged to take up careers like mathematics, she initially faced parental resistance to her mathematical pursuits. Her parents however gave in once they recognised her mathematical brilliance. Apprehending societal opposition she faked her identity as a man and took a pseudonym Monsieur le Blanc, a former student at Ecole Polytechnic. She managed to get materials meant for le Blanc and week after week submitted answers to the problems. Louis Joseph Lagrangee, himself one of the greatest mathematicians and supervisor at Ecole recognised the brilliance of the answers and asked for a meeting with Monsieur le Blanc. It was thus that Germain was obliged to reveal her identity. She got interested in number theory and quite inevitably came to know of the Fermat's Last Theorem. She decided to share her work with the top number theorist of the time Carl Friedrich Gauss. Again apprehending that Gauss may not take her seriously because of her gender she once again took to her previous pseudonym. By the time she had made considerable progress. But after sometime she lost contact with Gauss and gave up her pursuit of FLT and turned to physics and did remarkable work on elasticity theory. Her true identity was revealed to Gauss when she was able to persuade invading french generals to guarantee safety of Gauss. This is how Gauss renewed contact with her. The later mathematicians who worked on Fermat's Last Theorem built on the work of Sophie Germain. The primes which when multiplied by two and increased by one yield another prime are called Germain primes after her.

Search for Fermat's solution, in the meantime continues...

## Comments

**Jaspalkaler (author)** from Chandigarh, India on July 31, 2012:

Thank you Soumyasrajan for the nice comment. Yes, may be I was over enthusiastic while drawing conclusion from Gauss's comment. But let me assure you I have greatest respect for his intellect and his contributions to both mathematics and physics.

**soumyasrajan** from Mumbai India and often in USA on July 31, 2012:

Nice article- very nice one Jaspal. I specially enjoyed reading Marie Sophie Germain.

I am not so sure that one can say that Gauss had really shown contempt. I thought his sentiment was "I do not want to spend time on such questions as I can make many similar statements which can neither be proved or disproved easily" So sentiment was more of "I will rather work on problems in which I am interested".

Mathematics does have many statements of this type.

One minor point - did you know that Pythagoras theorem was known in India with proof a few hundred years earlier that Pythagoras.