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RAMANUJAMS NUMBER

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This article is about the number 1729. For the year, see 1729.

List of numbers — Integers ← 1k 2k 3k 4k 5k 6k 7k 8k 9k →
1729
Cardinal	One thousand seven hundred [and] twenty-nine
Ordinal	1729th
Factorization
Divisors	7, 13, 19, 91, 133, 247
Roman numeral	MDCCXXIX
Binary	11011000001
Octal	3301
Duodecimal	1001
Hexadecimal	6C1

1729 is known as the Hardy-Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to 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 quotation is sometimes expressed using the term "positive cubes", as the admission of negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a factor of 1729):

91 = 6³ + (−5)³ = 4³ + 3³

Of course, equating "smallest" with "most negative", as opposed to "closest to zero" gives rise to solutions like −91, −189, −1729, and further negative numbers. This ambiguity is eliminated by the term "positive cubes".

Numbers such as

1729 = 1³ + 12³ = 9³ + 10³

that are the smallest number that can be expressed as the sum of two cubes in n distinct ways have been dubbed taxicab numbers. 1729 is the second taxicab number (the first is 2 = 1³ + 1³). The number was also found in one of Ramanujan's notebooks dated years before the