The number 1,729 is not one to make the average person’s pulse race, but it is the subject of one of the most remarkable stories in the history of mathematics.
Most of us learnt basic arithmetic at school, and we all remember that some students were better at it than others – the bright girl who could do sums twice as fast as the rest of us, or the boy who could prove theorems in a trice. Of course all subjects attract a range of skills, but almost unique to mathematics are a handful of extreme outliers who are so good it seems they are deploying some form of magic. The best-known genius of this type was Srinivasa Ramanujan.
Born in 1887, Ramanujan was an eccentric young Indian student who lived in obscurity in the town of Kumbakonam in the state of Tamil Nadu. Bestowed with remarkable analytical skills, by the age of 13 he had devised his own scheme for computing the digits of pi that is still in use today. He spent much of his spare time scribbling formulae in notebooks or on a small blackboard.
By the age of 23 Ramanujan was convinced he was making important new discoveries in mathematics, and was enterprising enough to write a letter to the eminent Cambridge Professor of Mathematics G.H. Hardy. “I beg to introduce myself to you as a clerk in the accounts department of the Port of Madras,” he began. “I have had no university education.” Ramanujan then set out some of his remarkable results.
It is easy to imagine a distinguished professor such as Hardy shrugging aside this letter arriving out of the blue from an unknown amateur in faraway Madras. But to his great credit, Hardy recognised a touch of pure genius in Ramanujan’s theorems, many of which were highly unusual in their form and betrayed an extraordinary originality. And this although most of Ramanujan’s theorems were merely stated as fact, with no formal proof accompanying them. It was almost as if the young Indian had plucked the results ready-made from some abstract realm of mathematical forms and relationships. When Hardy replied asking about proofs, Ramanujan was coy, saying he had his own unusual methods and that, without proper explanation, “you will at once point me to the lunatic asylum”.
The young Indian set about working on hundreds of new theorems, dazzling his peers who were baffled as to the source of his extraordinary abilities.
Recognising that genius and eccentricity often go hand-in-hand, especially in mathematics, Hardy arranged to bring Ramanujan to England. But there were serious obstacles. As a devout Hindu and an orthodox Brahmin, travelling to a foreign land presented many cultural difficulties, not least in regard to his strict diet. After months of deliberation and consultation, Ramanujan finally decided to accept Hardy’s offer, and on 17 March, 1914 he set out by ship with some trepidation.
Once in Cambridge, the young Indian set about working on hundreds of new theorems, dazzling his peers who were baffled as to the source of his extraordinary abilities. Hardy said: “I have never met Ramanujan’s equal.” Because of his lack of formal education, Ramanujan was able to work at Cambridge only by being enrolled as a student at Trinity College.
Although he was now ensconced in the world centre of pure mathematics and was at last receiving the recognition he deserved, Ramanujan did not fare so well in his private life. His sensitive and unusual personality and strict dietary requirements proved deeply problematic. He had trouble obtaining the correct ingredients for his meals and his religion forbade him from eating with others in his Cambridge college. He became homesick and began to lose weight. His fragile health suffered, especially during the English winters. He even became suicidal.
Eventually Ramanujan was confined to a nursing home to await his return to India. Hardy paid frequent visits to his friend and colleague. Not surprisingly, the conversation usually turned to mathematics. On one such visit, 1,729 cropped up. This was the number of the taxi cab Hardy had taken to the clinic, and as befits two number theorists they discussed its significance. Hardy thought 1,729 to be a boring run-of-the-mill number, but Ramanujan disagreed. “That is a really, really interesting number,” he declared. How so? “It is the smallest number that can be expressed as the sum of two cubes in two different ways!”
Ramanujan could see immediately that: 123+ 13 = 103 + 93 = 1,729.
This amusing anecdote came to symbolise Ramanujan’s humble genius, and numbers that can be expressed as the sum of two cubes in two separate ways are known as “taxi numbers” in recognition. Other taxi numbers are 4,014, 13,832 and 20,638. But 1,729 is the smallest.
Sadly, Ramanujan never regained his health. He died on 20 April, 1920 in a care home near Madras (now Chennai). He continued working on new theorems even on his death bed. To this day nobody can say how Ramanujan came to have this incredible ability, but it is fascinating to speculate that there may be other Ramanujans out there, awaiting an enlightened mentor such as Hardy.
Paul Davies is Regents' Professor and Director of the Beyond Centre for Fundamental Concepts in Science at Arizona State University. He is also a prolific author, and Cosmos columnist.
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