# A Beautiful Ikigai

Well, there are ‘most’, ’best’, and ‘greatest’ things which human define and assign! We say, the highest mountain known in the world is Mt. Everest. I say, the best illustration of having power of intuition is Ramanujan! And many more such things. Similarly, I would like to point the most successful and beautiful ikigai that one would want to know…

What is it? It is ikigai of a small kid who once decided to solve Fermat’s last theorem. It is ikigai of a person who cracked the most difficult problem in the world. It is ikigai of the person who worked seven years secretly to achieve his childhood dream. It is the ikigai of him who has received 2016 Abel prize (some considers this as ‘Noble Of Mathematics’) for his same contribution. Yes, I am talking about Sir Andrew Wiles.

It is a story of child, a story started when Andrew was 10 years old. He, strolling in a public library came across a book ‘A last problem’ by E.T. Bell. The book is all about Fermat’s last theorem. He read the problem and searched for the proofs. But, there was no proof. It was a problem stated back three centuries, without a proof. A problem theorem which is quite simple to understand even for bright school going child had not proved yet! Since then, Andrew decided to find the missing proof of such an old problem. The problem, Fermat’s last theorem (FLT) is: ** x^n+y^n=z^n has no whole number solution for n≥3. **It is FLT because FLT is the last one (among problems stated by Pierre Fermat) that anyone can find poof for. It was unsolved for centuries. More the people tried, more it failed, and more it became beautiful. Fermat stated this problem along with *a marginal note* as “** Hanc marginis exiguitas non caperet, Demonstrationem mirabilem (I have a truly marvellous demonstration of this which this margin is too narrow to contain.)**” And a journey began. A journey of success, failure, triumph, jealousy, tragedy, intense pressure and competition! Fermat was unaware about that this one is going to be the most difficult and probably most discussed problem in the sphere of mathematical endeavour.

The whole journey of this beautiful problem is not a point now. The focus is Andrew and FLT.

Andrew did his higher education in mathematics. He pursued PhD in a field of number theory. But, still, FLT was there, unproved! Someday, in 1986, in his late 30’s, someone showed that there is a link between Taniyama Shimura Conjecture (TS) and FLT. And, all over again, he decided to try for it, he has to conquer TS, eventually, he has to get FLT, his childhood dream! His passion for the problem! He has to show all elliptic curves are modular (TS). He tried every tool and technique which was used in past. He developed new ones if needed.

*Instead what was required was a logical step-by-step argument which would effectively give a reason and explain why every elliptic equation had to be modular….. “I carried this thought around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day and I would be thinking about it when I went to sleep. Without distraction I would have the same thing going round and round in my mind.**” –*** Andrew Wiles** (A book -Fermat’s last theorem, by Simon Singh).

He enjoyed many breakthroughs momentarily like toppling first domino in domino effect (Proof by Induction) using Galois Theory! But, there were such many ‘nexts’ yet. There were some years when he knew he was at a right track but it could be that the tools needed to solve are beyond present day mathematics. “So, even if I was on right track, I could be living in wrong century.” He worked on other theories like Iwasawa theory. But, it was just failure in order to make progress in FLT. Then, another theory called ‘Kolyvagin-Flach method (K-F method, recent work in elliptic curves at the time) helped him to build the proof. Week after week, he was developing it, making the methods more powerful. Well, it was a time of 1991-92. Six years! The method worked for family of elliptic equations. But, a family of elliptic equations was still remaining. Last element of proof is to be proved yet!

Here it is:

** “I was casually looking at a paper of Barry Mazur’s and there was one sentence there that just caught my attention. It mentioned a nineteenth-century construction, and I suddenly realised that I should be able to use that to make the Kolyvagin–Flach method work on the final family of elliptic equations.” **(A book -Fermat’s last theorem, by Simon Singh)

And, it helped. FLT was proved. He gave a lecture series on three days to explain his work. Now it was a time for checking the proof and announcing officially that FLT is proved. The process of checking started. A nightmare mail arrived that there is slight problem in proof. There was a serious mistake in proof. So no FLT. A time, frustrating embarrassing time for Andrew! Is that like a work of seven years in complete secrecy goes in vain? Absolutely no, he developed new mathematics, we know learning it is not an easy task, imagine of creating it!

Andrew decided to give it a last try. “I couldn’t give up. I was obsessed by this problem and I still believed that the Kolyvagin–Flach method just needed a little tinkering. I just needed to modify it in some small way and then it would work just fine.” If it is impossible for him to provide the proof, at least, he thought to know why K-F method didn’t work. And, suddenly, he revealed that Iwasawa theory (which he tried before some years) and K-F method together is the key to proof. He worked on it. And the problem was solved finally after one year of the ‘slight mistake.’

*A problem worthy of attack,*

** Proves its worthy by fighting back!** (Piet Hein)

At last, the most difficult problem, stated way back in seventeenth century, Fermat’s Last Theorem was solved officially. Sir Andrew Wiles collected the Wolfskehl prize in 1997.

*“This was my childhood passion, dream. …Having solved this problem there’s certainly a sense of loss, but at the same time there is this tremendous sense of freedom. I was so obsessed by this problem that for eight years I was thinking about it all the time – when I woke up in the morning to when I went to sleep at night. That’s a long time to think about one thing. That particular odyssey is now over. My mind is at rest.”*

I just think this is the most appealing ikigai. Just awesome. A true meaning of ikigai, rather a definition!

Nice !!