## Ramanujan surprises again

Ramanujan’s manuscript. The representations of 1729 as the sum of two cubes appear in the bottom right corner. The equation expressing the near counter examples to Fermat’s last theorem appears further up: α^{3} + β^{3} = γ^{3} + (-1)^{n}. Image courtesy Trinity College library. Click here to see a larger image.

A box of manuscripts and three notebooks. That’s all that’s left of

the work of Srinivasa Ramanujan, an Indian mathematician who lived

his remarkable but short life around the beginning of the

twentieth century. Yet, that small stash of mathematical legacy still

yields surprises. Two mathematicians of Emory University, Ken

Ono and Sarah Trebat-Leder,

have recently made a fascinating discovery within its yellowed pages. It shows

that Ramanujan was further ahead of his time than anyone had

expected, and provides a beautiful link between several milestones in the history of

mathematics. And it all goes back to the innocuous-looking number 1729.

Ramanujan’s story is as inspiring as it is tragic. Born in 1887 in a small village around 400 km from

Madras (now Chennai), Ramanujan developed a passion for mathematics at a young age, but had

to pursue it mostly alone and in poverty. Until, in 1913, he decided to write a

letter to the famous Cambridge number theorist G.H. Hardy. Accustomed

to this early form of spam, Hardy might have been

forgiven for dispatching the highly

unorthodox letter straight to the bin. But he

didn’t. Recognising the author’s genius, Hardy invited Ramanujan to Cambridge,

where he arrived in 1914. Over the following years, Ramanujan more

than repaid Hardy’s faith in his talent, but suffered ill health due, in part, to the

grizzly English climate and food. Ramanujan returned to India in

1919, still feeble, and died

the following year, aged only 32. Hardy later described his collaboration with

Ramanujan as “the one romantic incident in my life”.

### The taxi-cab number

The romanticism rubbed off on the number 1729, which plays a

central role in the Hardy-Ramanujan story. “I remember once going to see [Ramanujan]

when he was ill at Putney,” Hardy wrote later. “I had ridden in taxi

cab number 1729 and remarked that the number seemed to me rather a

dull one, and that I hoped it was not an unfavourable omen. ‘No’, he

replied, ‘it is a very interesting number; it is the smallest number

expressible as the sum of two cubes in two different ways.'” What

Ramanujan meant is that

The anecdote gained the number 1729 fame in mathematical circles, but until

recently people believed its curious property was just another random

fact Ramanujan carried about in his brain — much like a train spotter

remembers train arrival times. What Ono and Trebat-Leder’s discovery

shows, however, is that it was just the tip of an ice berg. In reality

Ramanujan had been busy developing a theory that was several decades ahead

of its time and yields results that are interesting to mathematicians even today. He just didn’t live long enough to publish

it.

The discovery came when Ono and fellow mathematician Andrew Granville were

leafing through Ramanujan’s manuscripts, kept at the Wren Library at

Trinity College, Cambridge. “We were sitting right next to the

librarian’s desk, flipping page by page

through the Ramanujan box,” recalls Ono. “We came across this one

page which had on it the two representations of 1729 [as the sum of

cubes]. We started laughing immediately.”

### Fermat’s last theorem and near misses

Srinivasa Ramanujan (1887 – 1920).

or

or

and so on.

In 1637 the French mathematician Pierre

de Fermat confidently asserted that the answer is no. If is a whole number greater than then there are no positive whole number triples and such that

Fermat scribbled in the margin of a page in a book that he had “discovered a truly marvellous proof of this, which this margin is too narrow to contain”. Naturally, this assertion was like catnip to mathematicians, who subsequently drove themselves crazy, for over 350 years, trying to find this “truly marvellous proof”.

What the equation in Ramanujan’s manuscript illustrates is that Ramanujan had found a whole family (in fact an infinite family) of positive whole number triples and that very nearly, but not quite, satisfy Fermat’s famous equation for They are off only by plus or minus one, that is, either

or

Since any positive whole number triple satisfying the equation would render Fermat’s assertion (that there are no such triples) false, Ramanujan had pinned down an infinite family of near-misses of what would be *counter-examples* to Fermat’s last theorem.

“None of us had any idea that Ramanujan was thinking about anything

[remotely] related to Fermat’s last theorem,” says Ono. “But here on a page, staring

us in the face, were infinitely many near counter-examples to it, two

of which happen

to be related to 1729. We were floored.” Even today,

nearly 400 years after Fermat’s claim and 20 years after its

resolution, only a handful of mathematicians even know about the

family Ramanujan had come up with. “I’m a Ramanujan scholar and I wasn’t aware of

it,” says Ono. “Basically, nobody knew.”

### Elliptic curves and climbing K3.

But this isn’t all. When Ono and his graduate student Sarah

Trebat-Leder decided to investigate further, looking at other pages in

Ramanujan’s work, they found he had developed a sophisticated

mathematical theory that went beyond anything people had suspected. “Sarah and I spent time thinking more deeply about what Ramanujan had

really done, and it turns out that he anticipated [an area of] mathematic 30 or 40

years before anyone knew this field would exist. That’s what we are

excited about.”

It turns out that from looking at equations of the form

it’s not too large a mathematical step to considering equations of the form

where , and are constants. If you plot the points that satisfy such an equation (for given values of and ) in a coordinate system, you get a shape called an *elliptic curve* (the precise definition is slightly more

involved, see here). Elliptic

curves played an important role in the eventual proof of Fermat’s last theorem,

which was delivered in the 1990s by the mathematician Andrew

Wiles.

Ono and Trebat-Leder found that Ramanujan had also delved into the

theory of elliptic curves. He did not anticipate the path taken by

Wiles, but instead discovered an object that is more

complicated than elliptic curves. When objects of this kind were rediscovered around

forty years later they were adorned with the name of

*K3 surfaces* — in honour of the mathematicians Ernst

Kummer, Erich

Kähler and Kunihiko

Kodaira, and the mountain K2, which is as difficult to climb as K3

surfaces are difficult to handle mathematically.

That Ramanujan should have discovered and understood an exceedingly complicated K3 surface is in itself remarkable. But his work on the surface also provided an unexpected gift to Ono and Trebat-Leder, which links back to elliptic curves. Like all equations, any elliptic curve equation

naturally cries out for solutions: pairs of numbers that satisfy the equation. In the spirit of Fermat, you might look for whole number solutions, but number theorists usually give themselves a little more leeway. They look for solutions that are *rational numbers*, that is, numbers that can be written as fractions.

The elliptic curves corresponding to whole number values of a between -2 and 1 and whole number values of values of b between -1 and 2. Only the curve for a = b = 0 doesn’t qualify as an elliptic curve because it has a sharp corner.

Last year, in 2014, the

mathematician Manjul Bhargava won the Fields Medal, one of the highest

honours in mathematics, for major progress in this context. Bhargava

showed that most elliptic curves fall into one of two particularly simple classes. Either there are only finitely many rational number solutions; or there are infinitely many, but there is a recipe that produces all of them from just a single rational number solution.

(You can read our interview with Bhargava and our article exploring some of his work.)

If you sift through all elliptic curves in a systematic way, for example by ordering them according to the size of the constants and that appear in their formulas, then you are most likely only ever going to come across these “simple” elliptic curves. The probability of finding a more complicated one, which requires two or three solutions to generate them all, is zero. Searching for such elliptic curves systematically is like searching a haystack for a needle in a way that guarantees the needle will always slip through the net. To get at those more complicated elliptic curves, you need another method.

And this is exactly what Ramanujan came up with. His work on the K3 surface he

discovered provided Ono and Trebat-Leder

with a method to produce, not just one, but infinitely many elliptic

curves requiring two or three solutions to generate all other

solutions. It’s not the first method that has been found, but it required no effort. “We tied the world record on the problem [of finding such

elliptic curves], but we didn’t

have to do any heavy lifting,” says Ono. “We

did next to nothing, expcet recognise what Ramanujan did.”

### Physics and extra dimensions

There is another interesting twist to this story. While Ramanujan

was working in the abstract realms of number theory, physicists

studying real-world phenomena began developing the theory of quantum

mechanics. Although a triumph in its own right, it soon became clear

that the resulting quantum physics clashed with existing physical theories in an

unredeemable way. The rift still hasn’t been healed and

presents the biggest problem of twenty-first century physics (see here

to find out more). One attempt at rescuing the situation was the

development, started in the 1960s, of *string theory*, a prime

candidate for a “theory of everything” uniting the disparate strands

of modern physics.

G.H. Hardy (1877 – 1947).

A curious prediction of string theory is that the world we live in

consists of more than the three spatial dimensions we can see. The

extra dimensions, the ones we can’t see, are rolled up

tightly in tiny little spaces too small for us to perceive. The theory

dictates that those tiny little spaces have a particular geometric

structure. There’s a class of geometric objects, called

*Calabi-Yau manifolds*, which fits the bill (see this article to find out more). And

one of the

simplest classes of Calabi-Yau manifolds comes from, wait for it,

K3 surfaces, which Ramanujan was the first to discover.

Ramanujan could never have dreamt of this development, of course. “He was a whiz with formulas and I think

[his aim was] to construct those near counter-examples to Fermat’s

last theorem.” says Ono. “So he

developed a theory to find these

near misses, without recognising that the

machine he was building, those formulas that he was writing down,

would be useful for anyone, ever, in the future.”

Ono doesn’t rule out that Ramanujan’s manuscripts contain further

hidden treasures. “I’ve known about 1729 for thirty years. It’s a lovely, romantic

number. Ramanujan was a genius and we are still

learning about the extent to which his creativity led him to his

formulas. His work amounts to one box, kept at Trinity College, and

three notebooks, kept at the University of

Madras. That’s not a lot. It’s crazy that we are still figuring out

what he had in mind. When is it going to end?”

### Further reading

You can read more about the work of Ramanujan in *A disappearing number*. For experts, Ono and Trebat-Leder’s paper is available here.

### About this article

Ken

Ono is Asa Griggs Candler Professor of Mathematics and Computer Science at Emory University.

Sarah Trebat-Leder is a PhD student at Emory, where she is a Woodruff Fellow and NSF Graduate Fellow.

Marianne Freiberger is Editor of *Plus*. She interviewed Ono and Trebat-Leder in October 2015.

**Original Source**