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 Aiyangar 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 189:

189 = 6³+(-3)³ = 4³+5³

Of course, equating "smallest" with "most negative", as opposed to "closest to zero" gives rise to solutions like -189, -1729, and further negative numbers. This unclearness 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. The number was also found in one of his notebooks dated years before the incident.

1729 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 property -- the 1729th decimal place is the beginning of the first occurrence of all ten digits consecutively in the decimal representation of e, although, of course, this fact would have been unknown to either mathematician, since the computer algorithms used to discover this weren't 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.

Bender, the robot on the television show Futurama, has a serial number of 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." The serial-number on the ship Nimbus (which also is from Futurama), contains this magic number as well.

Quotations

• "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