Ramanujan’s gift: solutions to elliptic curves

1729 is the sum of two cubes in two different ways.

As numbers go, 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 elliptical curves.

The anecdote goes that once when Hardy visited Ramanujan who was sick, Hardy remarked: “I had ridden in taxicab 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 two different ways.”

Yes, 1729 = 93 + 103 and also 1729 = 123 +13

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 93 + 103 = 123 +13, scribbled in a corner. Recognising 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 equation,” 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 Trebat 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 recognised 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 Ono and Trebat Leder, it transpires that he knew of these functions long before. Dr Ono continues, “Ramanujan produced so many mysterious formulas, 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, “He [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 formulas immediately set the record on the problem. We hardly had any work to do. Ramanujan’s formula was a gift to us.”