# 1729 (number)

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] (http://www-gap.dcs.st-and.ac.uk/~history/Quotations/Hardy.html):

 List of numbers — Integers << 1k 2k 3k 4k 5k 6k 7k 8k 9k >>
1729
CardinalOne thousand seven hundred
[and] twenty-nine
Ordinal1729th
Factorization[itex]7 \cdot 13 \cdot 19[itex]
Divisors7,13,19,91,133,247
Roman numeralMDCCXXIX
Binary11011000001
I remember once going to see him when he was lying 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 quote 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:

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

which can be expressed as the sum of cubes in distinct ways have been dubbed taxicab numbers. 1729 is the first non-trivial taxicab number (technically, 1 is the first taxicab number). The number was also found in one of Ramanujan's notebooks dated years before the incident.

1729 is the third Carmichael number, and a Zeisel number.

It is a centered cube number, as well as 12-gonal, 24-gonal and 84-gonal number.

1729 has another interesting property: the 1729th decimal place is the beginning of the first occurrence of all ten digits consecutively in the decimal representation of the transcendental number e, although, of course, this fact would have been unknown to either mathematician, since the computer algorithms used to discover this were not implemented till much later. [2] (http://www.mathpages.com/home/kmath028.htm)

Because in base 10 the number 1729 is divisible by the sum of its digits, it is a Harshad number. It also has this property in octal and hexadecimal, but not in binary.

The television show Futurama contains a running joke about the Hardy-Ramanujan number. In one episode, the robot Bender receives a card labeled "SON 1729". Ken Keeler, a writer on the show with a Ph. D. in Applied Math, said that "that 'joke' alone is worth six years of grad school." In another episode, Bender's serial number is one of a pair of elegant taxicab numbers: his number is 9523 + (-951)3 = 2716057, while that of fellow robot Flexo is 1193 + 1193 = 3370318. (This datum is one of the pieces of evidence the episode uses to establish that Bender and Flexo are a pair of good-and-evil twins.) The starship Nimbus displays the hull registry number NC-1729, incidentally also a riff on the USS Enterprise's NCC-1701. Finally, the episode "The Farnsworth Parabox" contains a montage sequence where the heroes visit several parallel universes in rapid succession, one of which is labeled "Universe 1729".

Some reports say that the octal equivalent (3301) was the password to Xerox Parc's main computer.

## Quotation

• "Every positive integer is one of Ramanujan's personal friends."—J. E. Littlewood, on hearing of the taxicab incident.

## References

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy