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]:

 1729 Template:Numbers 1000 - 10000 Cardinal One thousand seven hundred [and] twenty-nine Ordinal 1729th Factorization $\displaystyle 7 \cdot 13 \cdot 19$ 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 (which is coincidentally 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 incident.

1729 is the third Carmichael number, and a Zeisel number. It is a centered cube number, as well as a dodecagonal number, a 24-gonal and 84-gonal number.

Investigating pairs of distinct integer-valued quadratic forms that represent every integer the same number of times, Schiemann found that such quadratic forms must be in four or more variables, and the least possible discriminant of a four-variable pair is 1729 (Guy 2004).

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.

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 until much later. [2]

Masahiko Fujiwara showed that 1729 is one of four natural numbers (the others are 81 and 1458 and the trivial case 1) which, when its digits are added together, produces a sum which, when multiplied by its reversed self, yields the original number:

1 + 7 + 2 + 9 = 19
19 · 91 = 1729

References to 1729

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, which simultaneously riffs 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".

The physicist Richard Feynman demonstrated his abilities at mental calculation when, during a trip to Brazil, he was challenged to a calculating contest against an experienced abacist. The abacist happened to challenge Feynman to compute the cube root of 1729.03; since Feynman knew that 1729 was equal to 123+1, he was able to give an accurate value for its cube root mentally using interpolation techniques (specifically, Binomial Expansion). The abacist had to solve the problem by a more laborious algorithmic method, and lost the competition to Feynman.

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

The play Proof by David Auburn also contains a reference to 1729.

Quotation

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