1729

The above pic from an acquaintance, had me immediately asking - New Car? 

A terse and brief NO! got me thinking. 

A discussion with my friend Umayal, made us zero in on the number rather than the car. 

Further googling enlightened me to the existence of the Ramanujan Hardy Number or what is popularly known as the Smallest Taxi Cab Number.

The Story behind the number 

The story of the number 1729 goes back to 1918 when Indian mathematician Srinivasa Ramanujan lay sick in a clinic near London and his friend and collaborator G.H. Hardy paid him a visit. 

Hardy said that he had arrived in taxi number 1729 and described the number “as rather a dull one." Ramanujan replied to that saying, “No, Hardy, it’s a very interesting number! It’s the smallest number expressible as the sum of two cubes in two different ways."

The Original British Taxi

The Number 1729

Ramanujan, in his ailing state saw that 1729 can be represented as

1³ + 12³ = 1 + 1,728 = 1,729 and

9³ + 10³ = 729 + 1,000 = 1,729

It is rare that a number can be split into two positive cubes, and even rarer that it can be split into two positive cubes in two different ways, and 1,729 is the smallest number that exhibits this property.

 

1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232  is the sequence of Taxi cab Numbers. (A001235)

Significance of the Number

But now mathematicians have discovered that there is more to 1729 than a casual conversation between Hardy and Ramanujan. Emory University researchers say that Ramanujan showed how the number is also related to elliptic curves and K3 surfaces—objects which play key roles today in string theory and quantum physics.

Ken Ono, a number theorist has declared that Ramanujan actually discovered a K3 surface more than 30 years before others started studying K3 surfaces and they were even named.

He further states that, Ramanujan’s work anticipated deep structures that have become fundamental objects in arithmetic geometry, number theory and physics.

Ramanujan's Deathbed Notes :

In 2013, while searching through Ramanujan archive at Cambridge, Ono unearthed a page of formulas that Ramanujan wrote a year after the 1729 conversation between him and Hardy. “From the bottom of one of the boxes in the archive, I pulled out one of Ramanujan’s deathbed notes," Ono recalls. “The page mentioned 1729 along with some notes about it." 

Ono and his graduate student Sarah Trebat-Leder published a paper about these new insights in the journal Research in Number Theory. The paper described how one of Ramanujan’s formulas associated with the taxi-cab numbers had unearthed secrets of elliptic curves. “We were able to tie the record for finding certain elliptic curves with an unexpected number of points, or solutions, without doing any heavy lifting at all," Ono explains. “Ramanujan’s formula, which he wrote on his deathbed in 1919, is that ingenious. It’s as though he left a magic key for the mathematicians of the future," Ono added.

Although Elliptic curves have been studied for many years, in the last 50 years they have been found to have an impact outside mathematics in areas such as Internet cryptography systems that protect information like bank account numbers.

What are K3 Surfaces?

André Weil named ‘K3 surfaces’ after the mathematicians Kummer, Kähler and Kodaira.

They are also named thus after the second highest mountain in the world, K2.

K3 surfaces are a certain class of smooth 2-dimensional complex varieties, hence the name ‘surfaces’. But they’re 4-dimensional when viewed as real manifolds. 

Alternately, a K3 surface is a simply connected Calabi–Yau manifold of complex dimension 2.

A smooth quartic surface in 3-space. 

The figure shows part of the real points (of real dimension 2) 

in a certain complex K3 surface (of complex dimension 2, hence real dimension 4).

 

 Plaster Model of a Kummer Surface now at the University of Oxford. designed by Karl Rohn.

Elliptic curves and K3 surfaces form an important next frontier in mathematics, and Ramanujan gave remarkable examples illustrating some of their features that was not known before. He identified a very special K3 surface, which belongs to a certain special family of elliptic curves. 

Ramanujan had found a K3 surface, much before they were officially identified and named by mathematician André Weil during the 1950s. Ramanujan was using 1729 and elliptic curves to develop formulas for a K3 surface, while mathematicians today, still struggle to manipulate and calculate with K3 surfaces. 

Enough about the number and its mathematical connotations.

Now to move onto the Story of Friendship between Hardy and Ramanujan - 

The year 1913 marked the beginning of an extraordinary relationship between an impoverished Indian clerk and a Cambridge don. 

GH Hardy (1877-1947) and Srinivasa Ramanujan (1887-1920) were the archetypal odd couple. Hardy, whose parents were both teachers, grew up in a middle-class home in Surrey, England. At the age of two he was writing numbers that reached into the millions, so it was no surprise that he eventually read mathematics at Trinity College, Cambridge, where he joined an elite secret society known as the Cambridge Apostles.

Ramanujan was born in the Indian state of Tamil Nadu. At the age of two he survived a bout of smallpox, but his three younger siblings were less fortunate, each one dying in infancy. Although he was enrolled in a local school, Ramanujan's most valuable education was thanks to a library book, A Synopsis of Elementary Results in Pure Mathematics by GS Carr, which contained thousands of theorems. He investigated these theorems one by one, relying on a chalk and slate for calculations, using his roughened elbows as erasers.

Aged 21, he married Janakiammal, who was just 10 years old. Unable to afford college fees and needing to support his wife, Ramanujan got a job as a clerk. Nevertheless, he continued his interest in mathematics in his spare time, developing novel ideas and proving fresh theorems.

Curious about the value of his research, Ramanujan began to write to mathematicians in England in the hope that someone would mentor him, or at least give him feedback. Academics such as MJM Hill, HF Baker and EW Hobson largely ignored Ramanujan's pleas for help, but Hardy was mesmerised by the two packages he received in 1913, which contained a total of 120 theorems.

Hardy's reaction veered between "fraud" and so brilliant that it was "scarcely possible to believe". In the end, he concluded that the theorems "must be true, because, if they were not true, no-one would have the imagination to invent them".

The British professor made arrangements for the young Indian, still only 26, to visit Cambridge. Hardy took great pride in being the man who had rescued such raw talent and would later call it "the one romantic incident in my life".

The resulting partnership gave rise to discoveries in several areas of mathematics and Ramanujan's genius was recognised in 1918 when he was elected as a fellow of the Royal Society.

Ramanujan's career was brilliant, but ended prematurely when he began to suffer from tuberculosis. He returned to India in 1919 and died the following year, aged 32.

G H Hardy and Ramanujan

Futurama - The Animated Series from Simpsons

Dr Ken Keeler, who swapped his job as a Mathematics Researcher to join the writing team behind the science fiction sitcom Futurama, is a great fan of Ramanujan.

He is one of the regular writers for The Simpsons and its sister series Futurama. Retaining his love for Ramanujan and his passion for numbers, he constantly smuggles mathematical references into both series.

In order to pay homage to Ramanujan, Keeler has repeatedly inserted 1,729 into Futurama.

• Bender, Futurama's cantankerous robot, has the unit number 1729.
• In an episode titled "The Farnsworth Parabox", the Futurama characters hop between multiple universes, and one of them is labelled "Universe 1729".
• The starship Nimbus has the hull registration number BP-1729.

This has certainly helped keep Ramanujan's memory alive, but it is probably not the sort of immortality that Hardy had in mind when he wrote: "Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. 

Immortality may be a silly word, but probably a mathematician has the best chance of whatever it may mean."

1729 - the Red Herrings!

Futurama Characters