Ramanujan Already Knew the Taxi's Number

G. H. Hardy took a cab to Putney, where Ramanujan was being treated for what was probably hepatic amoebiasis (long misdiagnosed as tuberculosis), and walked into the hospital room with nothing to say. Trying conversation, he mentioned the cab number — 1729 — and called it dull, hoping that wasn't a bad omen. Ramanujan replied that 1729 is, in fact, very interesting: it is the smallest number expressible as the sum of two positive cubes in two different ways.

The arithmetic checks out. 1³ + 12³ = 1 + 1728 = 1729. 9³ + 10³ = 729 + 1000 = 1729. Run the search by hand below 1729 and no smaller integer admits two such cube decompositions.

What the standard retelling skips is that Ramanujan didn't compute the answer in his hospital bed. Bruce Berndt, who edited his notebooks, has shown that the property of 1729 was already noted in Ramanujan's pre-Cambridge writing — he had been collecting curios about cubes and quartics for years. Hardy's remark was a cue, not a question.

The story did, however, name the concept. Numbers that can be written as a sum of two positive cubes in n different ways are now called taxicab numbers, denoted Ta(n). Ta(1) = 2 (just 1³ + 1³). Ta(2) = 1729. Ta(3) = 87,539,319, found by John Leech in 1957 — large enough that you would not stumble over it in a cab fleet. The existence of Ta(n) for every n was proved by Hardy and E. M. Wright in their 1938 textbook, but the actual values get found one at a time and the search is hard.

Hardy himself set the gloss in his memoir A Mathematician's Apology: Ramanujan, he wrote, could see relationships between numbers that to most mathematicians looked like strangers in a crowd. The 1729 anecdote stuck because it gave the abstraction a face, and a cab.