# Ramanujan surprises again | plus.maths.org

Ramanujan’s manuscript. The representations of 1729 because the sum of two cubes seem within the backside proper nook. The equation expressing the close to counter examples to Fermat’s final theorem seems additional up: α^{three} + β^{three} = γ^{three} + (-1)^{n}. Image courtesy Trinity College library. Click right here to see a bigger picture.

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

the work of Srinivasa Ramanujan, an Indian mathematician who lived

his outstanding however brief life across the starting of the

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

yields surprises. Two mathematicians of Emory University, Ken

Ono and Sarah Trebat-Leder,

have just lately made an interesting discovery inside its yellowed pages. It reveals

that Ramanujan was additional forward of his time than anybody had

anticipated, and supplies an exquisite hyperlink between a number of milestones within the historical past of

arithmetic. And all of it goes again to the innocuous-looking quantity 1729.

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

Madras (now Chennai), Ramanujan developed a ardour for arithmetic at a younger age, however had

to pursue it principally alone and in poverty. Until, in 1913, he determined to write down a

letter to the well-known Cambridge quantity theorist G.H. Hardy. Accustomed

to this early type of spam, Hardy might need been

forgiven for dispatching the extremely

unorthodox letter straight to the bin. But he

did not. Recognising the writer’s genius, Hardy invited Ramanujan to Cambridge,

the place he arrived in 1914. Over the next years, Ramanujan extra

than repaid Hardy’s religion in his expertise, however suffered unwell well being due, partially, to the

grizzly English local weather and meals. Ramanujan returned to India in

1919, nonetheless feeble, and died

the next yr, aged solely 32. Hardy later described his collaboration with

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

### The taxi-cab quantity

The romanticism rubbed off on the quantity 1729, which performs a

central function in the Hardy-Ramanujan story. “I bear in mind as soon as going to see [Ramanujan]
when he was unwell at Putney,” Hardy wrote later. “I had ridden in taxi

cab quantity 1729 and remarked that the quantity appeared to me moderately a

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

replied, ‘it’s a very attention-grabbing quantity; it’s the smallest quantity

expressible because the sum of two cubes in two other ways.'” What

Ramanujan meant is that

The anecdote gained the quantity 1729 fame in mathematical circles, however till

just lately folks believed its curious property was simply one other random

truth Ramanujan carried about in his mind — very like a practice spotter

remembers practice arrival occasions. What Ono and Trebat-Leder’s discovery

reveals, nonetheless, is that it was simply the tip of an ice berg. In actuality

Ramanujan had been busy growing a concept that was a number of a long time forward

of its time and yields outcomes which can be attention-grabbing to mathematicians even right now. He simply did not stay lengthy sufficient to publish

it.

The discovery got here when Ono and fellow mathematician Andrew Granville have been

leafing by Ramanujan’s manuscripts, saved on the Wren Library at

Trinity College, Cambridge. “We have been sitting proper subsequent to the

librarian’s desk, flipping web page by web page

by the Ramanujan field,” recalls Ono. “We got here throughout this one

web page which had on it the 2 representations of 1729 [as the sum of

cubes]. We began laughing instantly.”

### Fermat’s final theorem and close to misses

Srinivasa Ramanujan (1887 – 1920).

or

or

and so forth.

In 1637 the French mathematician Pierre

de Fermat confidently asserted that the reply is not any. If is a complete quantity larger than then there are not any constructive entire quantity triples and such that

Fermat scribbled within the margin of a web page in a e-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 loopy, for over 350 years, looking for this “truly marvellous proof”.

What the equation in Ramanujan’s manuscript illustrates is that Ramanujan had discovered a complete household (actually an infinite household) of constructive entire quantity triples and that very practically, however not fairly, fulfill Fermat’s well-known equation for They are off solely by plus or minus one, that’s, both

or

Since any constructive entire quantity triple satisfying the equation would render Fermat’s assertion (that there are not any such triples) false, Ramanujan had pinned down an infinite household of near-misses of what could be *counter-examples* to Fermat’s final theorem.

“None of us had any concept that Ramanujan was fascinated by something

[remotely] associated to Fermat’s final theorem,” says Ono. “But right here on a web page, staring

us within the face, have been infinitely many close to counter-examples to it, two

of which occur

to be associated to 1729. We have been floored.” Even right now,

practically 400 years after Fermat’s declare and 20 years after its

decision, solely a handful of mathematicians even know concerning the

household Ramanujan had give you. “I am a Ramanujan scholar and I wasn’t conscious of

it,” says Ono. “Basically, no one knew.”

### Elliptic curves and climbing K3.

But this is not all. When Ono and his graduate scholar Sarah

Trebat-Leder determined to research additional, different pages in

Ramanujan’s work, they discovered he had developed a classy

mathematical concept that went past something folks had suspected. “Sarah and I spent time pondering extra deeply about what Ramanujan had

actually performed, and it seems that he anticipated [an area of] mathematic 30 or 40

years earlier than anybody knew this subject would exist. That’s what we’re

enthusiastic about.”

It seems that from equations of the shape

it’s not too giant a mathematical step to contemplating equations of the shape

the place , *elliptic curve* (the exact definition is barely extra

concerned, see right here). Elliptic

curves performed an vital function within the eventual proof of Fermat’s final theorem,

which was delivered within the 1990s by the mathematician Andrew

Wiles.

Ono and Trebat-Leder discovered that Ramanujan had additionally delved into the

concept of elliptic curves. He didn’t anticipate the trail taken by

Wiles, however as an alternative found an object that’s extra

difficult than elliptic curves. When objects of this sort have been rediscovered round

forty years later they have been adorned with the identify of

*K3 surfaces* — in honour of the mathematicians Ernst

Kummer, Erich

Kähler and Kunihiko

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

surfaces are tough to deal with mathematically.

That Ramanujan ought to have found and understood an exceedingly difficult K3 floor is in itself outstanding. But his work on the floor additionally offered an surprising present to Ono and Trebat-Leder, which hyperlinks again to elliptic curves. Like all equations, any elliptic curve equation

naturally cries out for options: pairs of numbers *rational numbers*, that’s, numbers that may be written as fractions.

The elliptic curves similar to entire quantity values of a between -2 and 1 and entire quantity values of values of b between -1 and a pair of. Only the curve for a = b = zero would not qualify as an elliptic curve as a result of it has a pointy nook.

Last yr, in 2014, the

mathematician Manjul Bhargava gained the Fields Medal, one of many highest

honours in arithmetic, for main progress on this context. Bhargava

confirmed that almost all elliptic curves fall into one in every of two notably easy courses. Either there are solely finitely many rational quantity options; or there are infinitely many, however there’s a recipe that produces all of them from only a single rational quantity resolution.

(You can learn our interview with Bhargava and our article exploring a few of his work.)

If you sift by all elliptic curves in a scientific manner, for instance by ordering them in response to the dimensions of the constants

And that is precisely what Ramanujan got here up with. His work on the K3 floor he

found offered Ono and Trebat-Leder

with a way to supply, not only one, however infinitely many elliptic

curves requiring two or three options to generate all different

options. It’s not the primary methodology that has been discovered, however it required no effort. “We tied the world report on the issue [of finding such

elliptic curves], however we did not

should do any heavy lifting,” says Ono. “We

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

### Physics and additional dimensions

There is one other attention-grabbing twist to this story. While Ramanujan

was working within the summary realms of quantity concept, physicists

finding out real-world phenomena started growing the speculation of quantum

mechanics. Although a triumph in its personal proper, it quickly grew to become clear

that the ensuing quantum physics clashed with present bodily theories in an

unredeemable manner. The rift nonetheless hasn’t been healed and

presents the most important drawback of twenty-first century physics (see right here

to search out out extra). One try at rescuing the state of affairs was the

growth, began within the 1960s, of *string concept*, a chief

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

of recent physics.

G.H. Hardy (1877 – 1947).

A curious prediction of string concept is that the world we stay in

consists of greater than the three spatial dimensions we are able to see. The

further dimensions, those we won’t see, are rolled up

tightly in tiny little areas too small for us to understand. The concept

dictates that these tiny little areas have a specific geometric

construction. There’s a category of geometric objects, known as

*Calabi-Yau manifolds*, which inserts the invoice (see this text to search out out extra). And

one of many

easiest courses of Calabi-Yau manifolds comes from, watch for it,

K3 surfaces, which Ramanujan was the primary to find.

Ramanujan might by no means have dreamt of this growth, after all. “He was a whiz with formulation and I feel

[his aim was] to assemble these close to counter-examples to Fermat’s

final theorem.” says Ono. “So he

developed a concept to search out these

close to misses, with out recognising that the

machine he was constructing, these formulation that he was writing down,

could be helpful for anybody, ever, sooner or later.”

Ono would not rule out that Ramanujan’s manuscripts comprise additional

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

quantity. Ramanujan was a genius and we’re nonetheless

studying concerning the extent to which his creativity led him to his

formulation. His work quantities to at least one field, saved at Trinity College, and

three notebooks, saved on the University of

Madras. That’s not lots. It’s loopy that we’re nonetheless determining

what he had in thoughts. When is it going to finish?”

### Further studying

You can learn extra concerning the work of Ramanujan in *A disappearing quantity*. For consultants, Ono and Trebat-Leder’s paper is out there right here.

### About this text

Ken

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

Sarah Trebat-Leder is a PhD scholar at Emory, the place she is a Woodruff Fellow and NSF Graduate Fellow.

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