# 1729 (number)

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 1729 Cardinal One thousand seven hundred [and] twenty-nine Ordinal 1729th Factorization $7 \cdot 13 \cdot 19$ Divisors 1, 7, 13, 19, 91, 133, 247, 1729 Roman numeral MDCCXXIX Binary 11011000001 Octal 3301 Duodecimal 1001 Hexadecimal 6C1

1729 is the natural number following 1728 and preceding 1730. 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 incident.

The same expression defines 1729 as the first in the sequence of "Fermat near misses" (sequence A050794 in OEIS) defined as numbers of the form 1 + z3 which are also expressible as the sum of two other cubes.

1729 is the third Carmichael number and the first absolute Euler pseudoprime.

1729 is 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 (1729 = 33018, 3 + 3 + 0 + 1 = 7) and hexadecimal (1729 = 6C116, 6 + C + 1 = 1910), 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.[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 reversal, yields the original number:

1 + 7 + 2 + 9 = 19
19 · 91 = 1729

The proof is very easy, which is probably why Fujiwara has never shown his proof. It suffices only to check sums up to 36, since an n-digit sum, when multiplied by its reversal, results in a number with at most 2n digits whose digit sum is no greater than 18n. For n > 2, 18n has less than n digits, and for n = 2, 18n = 36. In addition, since reversals and digit sums do not affect mod 9 arithmetic, the square of the sum is congruent to the sum itself mod 9, so the sum must be congruent to 0 or 1 (mod 9)

It has occasionally been suggested that Hardy's story is apocryphal, on the grounds that he almost certainly would have been familiar with some of these features of the number.

## References to 1729

• The television show Futurama contains several jokes about the Hardy-Ramanujan number. In one episode, the robot Bender receives a Christmas card from the machine that built him labeled "Son #1729". Ken Keeler, a writer on the show with a Ph. D. in Applied Math, said "that 'joke' alone is worth six years of grad school." In another episode, Bender's serial number is revealed to be the sum of two cubes: his number is 2716057 = 9523 + (−951)3, while that of fellow robot Flexo is 3370318 = 1193 + 1193. (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 BP-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 universe where Fry, Leela and Bender are all giant rude talking bobbleheads). In the movie, "Bender's Big Score", the number of the taxi cab Fry takes home in the past is also the sum of two cubes.
• 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 movie Lucky Number Slevin also references the number 1729 in association with the character Nick Fisher.

## Quotation

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

## References

• Martin Gardner, Mathematical Puzzles and Diversions, 1959
• Richard K. Guy, Unsolved Problems in Number Theory, 2nd ed., Springer, 2004. D1 mentions the Hardy-Ramanujan number.