The Hardy-Ramanujan Number

“I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “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.” — G.H. Hardy

Left: One of few photographs of Ramanujan. Right: Ramanujan’s manuscript. The representations of 1729 as the sum of two cubes appear in the bottom right corner. Photo: Trinity College library.

The two different ways 1729 is expressible as the sum of two cubes are 1³ + 12³ and 9³ + 10³. The number has since become known as the Hardy-Ramanujan number, the second so-called “taxicab number”, defined as

Taxicab numbers
The smallest number that can be expressed as the sum of two cubes in n distinct ways.

So far, six taxicab numbers are known. They are:

Ta(1) = 2
= 1³ + 1³

Ta(2) = 1,729
= 1³ + 12³ = 9³ + 10³

Ta(3) = 87,539,319
= 167³ + 436³…