taxicab number (n.) a number that can expressed as the sum of two cubes in two different ways
We try to steer clear of mathematics most of the time here on HH, because—well, it’s numbers. But this week, the fact that numbers that can be expressed as the sum of two cubes in two different ways are called taxicabs was just too good a fact to ignore.
Taxicab numbers are rare, and only half a dozen or so of them have ever been identified. The first and smallest is 2, the sum of the reversible calculation 1³ + 1³. After that you won’t come across another until you reach 1,789, which is equal to both 1³ + 12³ and 9³ + 10³. The gaps between the taxicabs grow ever larger the higher you count, but the number of ways of expressing them increases too: so the third taxicab number—87,539,319—can be expressed in three different ways; the fourth—6,963,472,309,248—in four different ways; and the fifth—48,988,659,276,962,496—in five different ways. The sixth and largest yet discovered is 24,153,319,581,254,312,065,344 (that’s just over 24.1 sextillion, should you want to read it out), which was identified in 2003.
Nope, these numbers aren’t called taxicabs because once one goes by you have to wait ages for the next one to come along. In fact, as many of you pointed out on Twitter, there’s a much better story to tell here.
In the late 1910s, the British mathematician and Cambridge scholar GH Hardy paid a visit to his friend and collaborator, the Indian mathematician Srinivasa Ramanujan, in a hospital in London. Hardy had taken a cab to the hospital bearing the number 1789, and on arriving commented to Ramanujan that 1,789 seemed to him a “rather dull” number, that he hoped wasn’t an “unfavourable omen” for his friend’s treatment. “No,” Ramanujan replied, “it is a very interesting number: it is the smallest number expressible as the sum of two cubes in two different ways.”
Numbers sharing this quality ultimately became known as taxicabs, while 1,789 is now known as the Hardy–Ramanujan number—and is just one of a number (no pun intended) of mathematical discoveries now bearing the duo’s names.
Hardy and Ramanujan first began working together in 1913, when Ramanujan sent nine pages of mind-boggling handwritten mathematical formulae from India to Hardy’s office in Cambridge. Impressed by Ramanujan’s obvious ability, Hardy arranged for him to come to England, where the pair soon began several years of highly productive collaboration.
Tragically, Ramanujan died in 1920 at the age of just 32. Hardy was later asked, after a lifetime of achievement and mathematical accomplishment, what he felt his own greatest contribution to mathematics was. Discovering Ramanujan was his immediate reply.