**M. Ram Murty ***obtained his PhD in 1980 from the Massachusetts Institute of Technology (MIT) specializing in number theory. *

*After post-doctoral fellowships at the Institute for Advanced Study in Princeton and the Tata Institute of Fundamental Research (TIFR) in Mumbai, he joined the faculty of McGill University in Montreal, Canada in 1982. *

*In 1990, he was elected Fellow of the Royal Society of Canada and in 1996 was awarded the Balaguer Prize (along with his brother Kumar Murty) for the monograph ‘Non-vanishing of L-functions and applications’ published by Birkhauser-Verlag. *

*In 1996, he moved to Queen's University where he now holds the Queen's Research Chair in Mathematics and Philosophy. In 2008, he was elected Fellow of the Indian National Science Academy and in 2012, Fellow of the American Mathematical Society. *

*He has written more than 10 books and over 150 research papers in mathematics. He spoke to* **Shubhashree Desikan**, while in Chennai recently.

**After the solution of Fermat’s last theorem what has been happening in mathematics?**

Unsolved problems are useful because they show us what’s missing in mathematics. The reason a problem in mathematics is unsolved is because we do not have the necessary tools or concepts. And the problem becomes the occasion for us to discover these new tools and concepts. In the case of Fermat’s last theorem (FLT) it opened up a whole collection of new concepts linking representation theory and number theory. And out of that emerged a powerful tool and this tool has become instrumental in solving many other questions that the lay public doesn’t know about. At least half a dozen major unsolved problems have been settled by the techniques that emerged from the solution of Fermat’s last theorem. Two examples are given by the Serre conjecture and the Sato-Tate conjecture. In fact, one of the major contributors to this theory is an Indian by the name Chandrashekhar Khare, who won the Cole Prize that is awarded by the American Mathematical Society for outstanding contributions in number theory. He is one the people who developed and pushed the new methods to greater heights.

**Will you comment on the primality proof of Agarwal, Kayal and Saxena?**

Well the interesting thing about the AKS discovery is that the mathematics used to solve an ancient longstanding problem is practically at the level of high-school algebra. Which means sometimes even the experts can be stumped into thinking that some new ideas are needed to solve some problems. I think it was a tremendous act of courage on the part of these three Indian mathematicians to have faith in their idea – that perhaps some elementary way of approaching the problem will work. They pushed that faith and I think they surprised and baffled even the experts.

**About your work...**

My work does not relate to Fermat’s last theorem. Some of the work that I’ve done recently and spoke about at the conference on Ramanujan in Delhi last week, has to do with new techniques that emanated from the solution of Fermat’s last theorem to another problem called the Sato-Tate conjecture. That conjecture more or less made predictions beyond that of Ramanujan. Ramanujan made some predictions on the Tau function and Sato and Tate conjecture made further predictions beyond that. Ramanujan’s three conjectures on the Tau function have been settled. (as of 1976) ...What I have done is to use the Sato-Tate conjecture, now that it is a theorem, to go further and see what are it’s applications and consequences.

Kumar (Kumar Murty) and I have just finished a book called the Mathematical Legacy of Srinivasa Ramanujan. The purpose of writing this book was to introduce to undergraduate students of mathematics some of Ramanujan’s work and the developments that came after that. That might be useful for an undergraduate level course or a graduate level seminar.

**Can Number Theory be used to popularise mathematics?**

The answer to that question is a resounding “Yes” because Number theory is a very beautiful topic in that the unsolved problems are easy to state in ways that a high school student can understand them.

Here is an example of an unsolved problem. Everyone knows that there are infinitely many prime numbers. Prime numbers are like 2,3,5,7,,11, 13 and so forth, numbers which do not have any proper divisors.

We know that there are infinitely many primes, p. The question arises whether p+2 is also a prime for infinitely many p. Such primes are called twin primes. For example, 3 is a prime and 5 is also a prime; similarly 5 is a prime and 7 is also a prime. But 7 and 9 are not both primes. 11 and 13 are twin primes. 17 and 19 are twin primes.

So you have these pairs of primes that differ by two. The question arises whether there are infinitely many such primes.

The interesting thing is that the question whether there are infinitely many such twin primes, is an open question. That’s a question I can explain to a high school student.

That’s an unsolved problem. We do not know the answer. We do know, however, that if p is a prime number, for infinitely many primes, p+2 is either a prime or a number that is divisible by at most two prime numbers. You see, now I have been able to explain a major achievement of twentieth century mathematics to a high school student. If students want to know how this comes about, then I will have to tell them that either they will have to take a course in number theory, or, (laughs) if you are Ramanujan, you go to a local library and teach yourself all of undergraduate mathematics.

So certainly number theory can be used ... to attract students to take up higher mathematics. I see that in Chennai the Pie Mathematics Association is using number theory and Ramanujan's life to popularise mathematics.

**Applied math can be understood and appreciated by many people while pure math is considered very elitist. Your comments.**

There are lots of examples where the purest of pure mathematics can be applied in a profound way. When Einstein came up with the theory of relativity, he needed something called Tensor Calculus and differential geometry.

Now these were developed in the 18th and 19th centuries purely for their own sake. These were mathematical theories and were beautiful, coherent and consistent.

Of course they had some applications, but not as dramatic as its use in theory of relativity. But the theory wouldn’t have been in place had someone worked it out with an application in mind.

*The copy has been re-edited for clarity*

## Sri Utpal Chattopadyay is right. It would appear, there would be a number of instances of p+2, such as 105 being a product of more than 2 prime numbers. I wrote on the same day to the Editor , The Hindu pointing out that the number depicting this year, namely 2013 is a product of three prime numbers, 3.11 and 61. The letter has not been published the past two days,but hopefully the contents and Sri Chattopadhyay's comment have been passed on to Ms, Shubashree Desikan and Dr. Ram Murty

## I am puzzled by the following statement the article : "We do know however that if p is a prime number, p+2 is either a prime or a number that is divisible by at most two prime numbers." How about 103, and 105? 103 is a prime. 105 is not a prime but 105 has three distinct prime divisors 3, 5, and 7. !

## I am no mathematician but have a tremendous love for it.

Reading this article I am attracted to the general idea of problems in mathematics. This

general idea has immense relevance to life. Hence I have slightly modified the statement of

Dr. Ram Murhty: "Unsolved problems are useful because they show us what is missing. The

reason a problem is unsolved is because we do not have the necessary tools or concepts.

And the problem becomes the occasion for us to discover these new tools and concepts."

The philosophical content of Dr. Ram Murthy is unmistakable.

Thus viewed the statement of Dr. Ram Murthy becomes a general lesson in life.

## Help to understand the importance of mathametics in life.

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