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

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]:

1729
Cardinal	One thousand seven hundred [and] twenty-nine
Ordinal	1729th
Factorization
Divisors	7,13,19,91,133,247
Roman numeral	MDCCXXIX
Binary	11011000001
Hexadecimal	6C1

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 second 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]

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".

Quotation

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

See also

• Interesting number paradox
• Berry paradox

References

• Martin Gardner, Mathematical Puzzles and Diversions, 1959
• The Dullness of 1729

External links

• MathWorld: Hardy-Ramanujan Number
• MathWorld: Taxicab Number