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."
Ramanujan, in his ailing state saw that 1729 can be represented as:
1³ + 12³ = 1 + 1,728 = 1,729
9³ + 10³ = 729 + 1,000 = 1,729
Because of this incident, 1729 is now known as the Ramanujan-Hardy number. To date, only six taxi-cab numbers have been discovered that share the properties of 1729. These are the smallest numbers which are the sum of cubes in different ways.
In fact, on Ramanujan’s 125th birth anniversary, a Mint columnist paid tribute to the mathematical genius by finding quirky properties of the number 125.
But now mathematicians have discovered that there is more to 1729 that 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.
“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," says Ken Ono, a number theorist at Emory in a public release. “It turns out that Ramanujan’s work anticipated deep structures that have become fundamental objects in arithmetic geometry, number theory and physics," added Ono.
In 2013, Ono searched through the Ramanujan archive at Cambridge and 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," said Ono.
Ono and his graduate student Sarah Trebat-Leder are publishing a paper about these new insights in the journal Research in Number Theory. The paper will describe how one of Ramanujan’s formulas associated with the taxi-cab number can unearth 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.
“This paper adds yet another truly beautiful story to the list of spectacular recent discoveries involving Ramanujan’s notebooks," says Manjul Bhargava, a number theorist at Princeton University. “Elliptic curves and K3 surfaces form an important next frontier in mathematics, and Ramanujan gave remarkable examples illustrating some of their features that we didn’t know before. He identified a very special K3 surface, which we can use to understand a certain special family of elliptic curves. These new examples and insights are certain to spawn further work that will take mathematics forward," Bhargava added in the public release by Emory University.
Ono had worked with K3 surfaces before and discovered that 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," Ono said. “Mathematicians today still struggle to manipulate and calculate with K3 surfaces. So it comes as a major surprise that Ramanujan had this intuition all along," Ono added.