## Tag Archives: Srinivas Ramanujan

### Genius of Srinivasa Ramanujan

1. In December 1914, Ramanujan was asked by his friend P.C. Mahalanobis to solve a puzzle that appeared in Strand magazine as “Puzzles at a Village Inn”. The puzzle stated that n houses on one side of the street are numbered sequentially starting from 1. The sum of the house numbers on the left of a particular house having the number m, equals that of the houses on the right of this particular house. It is given that n lies between 50 and 500 and one has to determine the values of m and n. Ramanujan immediately rattled out a continued fraction generating all possible values of m without having any restriction on the values of n. List the first five values of m and n.
2. Ramanujan had posed the following problem in a journal: $\sqrt{1+2\sqrt{1+3\sqrt{\ldots}}}=x$, find x. Without receiving an answer from the readers, after three months he gave answer as 3. This he could say because he had an earlier general result stating $1+x=\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+\ldots}}}}$ is true for all x. Prove this result, then $x=2$ will give the answer to Ramanujan’s problem.

Try try until you succeed!!

Nalin Pithwa.

### Ramanujan’s gift: solutions to elliptic curves

(The following article appeared in The Hindu, Nov 30, 2015 authored by Shubashree Desikan)

As numbers go by, 1729, the Hardy-Ramanujan number is not new to math enthusiasts. But, now this number has triggered a major discovery — on Ramanujan and the theory of what are known as elliptic curves.

The anecdote goes that once when Hardy visited Ramanujan who was sick, Hardy remarked: “I had ridden in taxivab number 1729, and it seems to me a rather dull number. I hope it was not an unfavourable omen.” To this, Ramanujan had replied, “No, it is a very interesting number. it is the smallest number expressible as the sum of two cubes in tow different ways.”

“Yes, $1729=9^{3}+10^{3}=12^{3}+1^{3}$ .”

This  story is often narrated to explain Ramanujan’s familiarity with numbers but not more than that. Recent discoveries have brought to light that it was far from coincidence that Ramanujan knew the properties of 1729. There are now indications that he had, in fact, been looking at more general structures of which this number was but an example.

Mathematicians Ken Ono and Andrew Granville were leafing through Ramanujan’s manuscripts at the Wren Library in Cambridge University, two years ago, when they came across the equation $9^{3}+10^{3}=12^{3}+1^{3}$, scribbled in a corner. Recognizing the representations of the number 1729, they were amused at first; then they looked again and found that there was another equation on the same page that indicated Ramanujan had been working even then, on a famous seventeenth century problem known as Fermat’s Last Theorem (proved by Andrew Wiles in 1994).

“I thought I knew all of the papers there, but to my surprise,we found one page with near misses to the Fermat equations,” writes Dr. Ono, who is also a Ramanujan scholar in an email to this correspondent. Having a sneaking suspicion that Ramanujan had a secret method that gave him his amazing formulas, Dr. Ono returned to Emory University and started working on these leads with his PhD student Sarah Trebet Leder.

“Together, we worked backwards through Ramanujan’s notes, and we figured out his secret. …[Ramanujan] arrived at the formulae on this page by producing a much more general identity. One which I recognized as a K3 surface ( a concept that mathematician Andrew Wiles used for solving Fermat’s Last Theorem), an object that mathematicians did not discover until the 1960s,” notes Dr. Ono.

Ramanujan died in 1920, long before mathematicians discovered the K3 surfaces, but from research done by Dr. Ono and Trebat Leder, it transpires that he knew these functions long before. Dr. Ono continues, ” Ramanujan produced so many mysterious formulae, which can be misunderstood at first glance. We have come to learn that Ramanujan was perhaps the greatest anticipator of mathematics. His bizarre methods and formulas have repeatedly offered hints of the future in mathematics. In this case, we have added to Ramanujan’s legend.”

Commenting on their own work on this, he says, “Ramanujan anticipated the theory of K3 surfaces before anyone had the merest glimpse. These surfaces are now at the forefront of research in mathematics and physics. In addition to adding to Ramanujan’s legacy Sarah and I were able to apply his formulas to a problem in number theory (finding large rank elliptic curves) and his formulae immediately set the record on the problem. We hardly had any work to do. Ramanujan’s formula was a gift to us.”

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Nalin Pithwa

### The man and his math by Sangeetha Devi Dundoo

The Hindu, 22 March 2011.

“[Srinivasa Ramanujan] was a genius far ahead of his times and math was his creative expression of numbers and not his means to make money. His was an extraordinary, inspiring and emotional journey,” says film director Dev Benegal. The filmmaker is interviewed about a film he plans to make on the life of mathematician Ramanujan, who grew up in poverty in Kumbakonam, India. Benegal, who has been researching Ramanujan’s life for four years, says “the film will explore his life at an emotional level–the struggle of his parents, particularly his mother; Ramanujan’s relationship with his wife, which is one of the greatest love stories of our times; the sacrifices that the wife had to make which are unknown and unheard of; and the bond that Ramanujan and G.H. Hardy shared.”

I like to inspire budding young minds towards Math…of course, you can say that I am biased, but …some people need inspiration, others have intrinsic motivation…I will share more such stories with you all later…

More later…

Nalin Pithwa