Ramanujan's Taxi Number

The mathematician G. H. Hardy once visited Ramanujan in the hospital. He mentioned he'd arrived in a taxi numbered 1729 and remarked it seemed like a rather dull number. As the story goes, Ramanujan immediately said, "No, it's actually very interesting. It's the smallest number expressible as the sum of two cubes in two different ways. It's 1 cubed plus 12 cubed. It's also 9 cubed plus 10 cubed. " Now, I want you to just think about that. He was lying in a hospital bed and he instantly, as the story goes, knew this property of 1729. No calculator, no pause, he just knew it. Which is why 1729 is now called the Hardy-Ramanujan number or the first taxicab number. The next taxicab number, the smallest expressible as a sum of two cubes in three different ways, is 87,539,319. That one took a bit longer to find. Littlewood once said that every positive integer was one of Ramanujan's personal friends. And 1729 is a pretty good example of why.