# Why is 1729 known as the Ramanujan number?

1729 is the natural number following 1728 and preceding 1730. It is known as the Hardy–Ramanujan number, after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. He related their conversation:

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 unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

The two different ways are:

1729 = 13 + 123 = 93 + 103

The quotation is sometimes expressed using the term “positive cubes”, since allowing negative perfect cubes (the cube of a negative integer) gives the smallest solution as
91(which is a divisor of 1729):

91 = 63 + (−5)3 = 43 + 33

Numbers that are the smallest number that can be expressed as the sum of two cubes in n distinct ways have been dubbed “taxi cab numbers “. The number was also found in one of Ramanujan’s notebooks dated years before the incident, and was noted by Frenicle de Bessy in 1657. A commemorative plaque now appears at the site of the Hardy-Ramanujan incident, 2 Colinette Road, Putney.

More stories to refer to in the chapter 5.3 from Chalk to Talk The Art of Teaching