Taxi-cab numbers, among the most beloved integers in math, trace their origins to 1918 and what seemed like a casual insight by the Indian genius Srinivasa Ramanujan. Now mathematicians at Emory University have discovered that Ramanujan did not just identify the first taxi-cab number – 1729 – and its quirky properties. He showed how the number relates to elliptic curves and K3 surfaces – objects important 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. “It turns out that Ramanujan’s work anticipated deep structures that have become fundamental objects in arithmetic geometry, number theory and physics.”
Ono and his graduate student Sarah Trebat-Leder are publishing a paper about these new insights in the journal Research in Number Theory. Their paper also demonstrates how one of Ramanujan’s formulas associated with the taxi-cab number can reveal 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 says. “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. All we had to do was recognize the key’s power and use it to drive solutions in a modern context.”