# Re-Examination Of Old Notebooks Reveals Powerful Mathematics

A bust of Ramanujan from the Birla Industrial and Technological Museum. Almost a century after his death Ramanujan is still changing mathematics. Credit: AshLin via Wikimedia Commons.

One of the most famous stories in pure mathematics has gained an extra chapter. The astonishing mathematical genius Srinivasa Ramanujan discovered fields of mathematics that have since proven invaluable to physicists, 30 years before the rest of the world caught up. Some of Ramanujan's work had not been replicated prior to a re-examination of his notebooks, and its rediscovery may be of assistance to physicists and cryptographers.

Ramanujan was a self-taught Indian prodigy who in a few years before his premature death produced arguably the most astonishing output in the history of pure mathematics. For decades mathematicians trawled his notebooks to find the nuggets of gold he left behind. However, according to Emory University's Ken Ono and Sarah Trebat-Leder, publishing in Research in Number Theory, some treasures were missed.

The new finds relate to the story non-mathematicians are most likely to know about Ramanujan, his invention of taxi-cab numbers. When Ramanujan was ill G.H. Hardy, the British mathematician who had discovered Ramanujan's brilliance, visited him in hospital. Hardy commented that the taxi-cab he had caught bore the number 1729, and he thought it a dull number. “No,” Hardy quotes Ramanujan as replying immediately, “It is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

Ramanujan had realized that 1729 is equal to both 93+103 and 123+13. Numbers that can be expressed by the formula a3+b3=c3+d3 have been known as taxi-cab numbers ever since, and a field of mathematics has opened up finding higher order versions, such as 87539319, the smallest number that is equal to the sum of two cubes in three different ways.

The number 1729 appears frequently in works written by mathematicians, most notably the series Futurama in one episode of which the number 87539319 appears on a taxi.

These obscure pursuits often bring complaints as to why tax-payer's money is spent to employ people in the hunt, but Ono and Trebat-Leder have an answer. Ramanujan invented a formula for finding examples of such numbers, using 1729 as one example. Re-examining Ramanujan's notes, Ono and Trebat-Leder find his formula describes elliptic curves and a type of smooth forms that are now known as K3 surfaces.

Elliptic curves and K3 surfaces have turned out to be powerful methods for understanding string theory and quantum mechanics. "We've found that Ramanujan actually discovered a K3 surface more than 30 years before others started studying K3 surfaces and they were even named," said Ono in a statement." It turns out that Ramanujan's work anticipated deep structures that have become fundamental objects in arithmetic geometry, number theory and physics."

The pair used Ramanujan's formula to seek examples of a particular category of elliptic curve, and were able to match the record without, in Ono's words, “doing any heavy lifting at all.”

The publication is excellently timed publicity for the forthcoming biopic of Ramanujan, on which Ono is an associate producer.

Ono said, “It's as though he left a magic key for the mathematicians of the future. All we had to do was recognize the key's power and use it to drive solutions in a modern context."