03 May 2009

Music of the non-primes

This is for you, Kate. The superscript 3 for cubes just showed up as the number 3, so I've put it in parentheses to make the equations less confusing.

April 22, 2009

Sexy maths: what's unique about the number 1,729

The number 1,729 has been appearing in some curious places recently. In the animated sitcom Futurama Bender was the 1,729th robot to be manufactured by its creator. The Nimbus BP-1729, the space craft captained by Zapp Brannigan, also pays homage to the number. The Complicite's hit show A Disappearing Number, features a man obsessed with getting his telephone number changed to include the digits 1,7,2 and 9. And in David Auburn's play Proof, the protagonist Catherine calculates at the beginning of the play that if every day she'd lost to depression was a year it would work out at a total of 1,729 weeks.

The thread that runs through all these strange occurrences of the number 1,729 is that the scripts were written by authors obsessed with mathematics. Because 1,729 isn't any old number but has some very interesting mathematical properties.

Not that every mathematician thought so. Indeed, the reason that 1,729 has such resonance for those obsessed with mathematics is because the great Cambridge mathematician G.H. Hardy once famously declared that he thought it a rather dull number. He said this while visiting his great collaborator Srinivasa Ramanujan at a London nursing home.

Hardy had discovered this untrained Indian genius some years earlier in 1913 after Ramanujan, a Hindu clerk at the Port Authority in Madras, had sent the Cambridge mathematician letters full of wild and unimaginable formulas. Hardy immediately arranged for Ramanujan to be brought to Cambridge where they'd worked together proving amazing theorems. But Cambridge couldn't accommodate Ramanujan's Brahmin customs. He had been used in India to someone handing him food as he calculated away. Suffering malnutrition, he fell gravely ill and depressed and eventually found himself confined to the nursing home in Putney.

Hardy sat next to his sick friend. But, being mathematicians, both were hopeless at small talk. So Hardy ventured 1,729, the number of the taxi he'd arrived in, as an example of a rather dull number.

Ramanujan's eyes lit up. “No, Hardy,” he replied. “It is a very interesting number. It is the smallest number expressible as the sum of two positive cubes in two different ways.”

He was right. Many numbers can be written as two cube numbers added together. For example 35=2(3)+3(3). But 1,729 is the smallest number that has two different ways that it can be split into cube numbers. One way is to write it as 12(3)+1(3) =1,728+1. But you can also express 1,729 as 10(3)+9(3)=1,000+729. And it was Ramanujan's extraordinary ability to recognise the special character of this number that sealed its place in mathematical folklore.

The story is frequently told to illustrate Ramanujan's special mathematical mind. He would often say that his mathematical discoveries came to him in dreams delivered by his family goddess Namagiri. A colleague of Hardy's in Cambridge said that Ramanujan seemed to know every number as if it were an intimate personal friend. But for me this is not the sign of a strange autistic mind, but an indication that Ramanujan was thinking about some of the deep problems that have fascinated mathematicians for millennia. Because the story is related to one of the central topics of mathematics: solving equations.

Take any number N and consider the equation x(3)+y(3)=N. These are examples of some of the most mysterious equations in mathematics, called elliptic curves.

One of the holy grails of mathematics is a problem called The Birch-Swinnerton- Dyer Conjecture, which tries to understand whether equations such as these have solutions or not. So important is the conjecture that there is a $1 million prize for the first person to crack it. But it isn't just important for mathematics. Some of the cutting-edge codes being used in industry exploit properties of these equations. Indeed, air-traffic control uses codes based on elliptic curves to keep information about flight paths secure from hackers. The number 1,729 is just the tip of one of the most mysterious topics in mathematics.

So next time you take a cab, spend the journey trying to unlock the interesting properties behind your taxi's number. As Ramanujan revealed to Hardy, there is no such thing as a dull number.


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