1729 (number)

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 visit to the hospital to see the Indian mathematician Srinivasa Ramanujan. In Hardy's words:^[1][2][3]

The two different ways are these:

1729 = 1³ + 12³ = 9³ + 10³

The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a divisor 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 that are the smallest number that can be expressed as the sum of two cubes in n distinct ways^[4] have been dubbed "taxicab numbers". The number was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by Frénicle de Bessy in 1657.

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.

Other properties

1729 is also the third Carmichael number and the first absolute Euler pseudoprime. It is also a sphenic number.

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 mildly 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.^[5]

Masahiko Fujiwara showed that 1729 is one of four positive integers (with the others being 81, 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

It suffices only to check sums congruent to 0 or 1 (mod 9) up to 19.

References to 1729

• The television show Futurama contains several jokes about the Hardy–Ramanujan number.^[2] 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 mathematics, 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).^[2] In the film, "Bender's Big Score", the number of the taxi cab Fry takes home in the past is 87539319, which is the third taxicab number, being the smallest number expressible as the sum of two cubes in three different ways.^[6]
• 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 (because one cubic foot equals 1728 cubic inches), he was able to mentally compute an accurate value for its cube root 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. The anecdote is related by Feynman in his memoir, Surely You're Joking, Mr. Feynman!.^[7]
• Some reports say that the octal equivalent (3301) was the password to Xerox PARC's main computer.
• The play Proof (and its adapted film) by David Auburn also contains a reference to 1729.
• The movie Lucky Number Slevin also references the number 1729 in association with the character Nick Fisher.
• The 2007 play A Disappearing Number by the Théâtre de Complicité company references the number. One of the main characters, Ruth, is a mathematician and 1729 are the last four digits of her phone number, paying homage to two of her heroes: Ramanujan and Hardy.

Quotation

• "Every positive integer is one of Ramanujan's personal friends."—J. E. Littlewood, upon hearing of the taxicab incident.^[8]

See also

• Taxicab number
• Interesting number paradox
• Berry paradox
• A Disappearing Number, a 2007 play about Ramanujan in England during World War I.

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.

Notes

External links

• MathWorld: Hardy–Ramanujan Number
• BBC: A Further Five Numbers
• Grime, James; Bowley, Roger. "1729: Taxi Cab Number or Hardy-Ramanujan Number". Numberphile. Brady Haran. 
• Why does the number 1729 show up in so many Futurama episodes?, io9.com