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Why is the Hardy-Ramanujan number 1729 special? What is the story behind it?

this is the smallest number that can be written as sum of two cubes in two different ways.

1729 = 1 3 + 12 3 = 9 3 + 10 3

the story

once  g h hardy came to visit ramanujan in hostipat in taxi number 1729. he said to ramanujan that is very dull and boring number. but ramanujan suddently said it is quite interesting. he said that is the smallest no that can be written as sum of two cubes in two different ways

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1729 is the natural number following 1728 and preceding 1730. 1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a visit to the hospital to see the Indian mathematician Srinivasa Ramanujan. In Hardy's words:[1]

“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," 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 these:

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
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INTERESTING.............!

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