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StarMountainKid's Commentary Blog

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The Hardy-Ramanujan Taxicab Number

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The British mathematician G.H. Hardy went to a hospital to visit the Indian mathematician Srinivasa Ramanujan: Hardy reported, “I had ridden in taxi cab number 1729, and remarked that the number seemed to me rather a dull one." "No," Ramanujan replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

Other properties of 1729:

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.

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

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.


The television show Futurama contains several jokes about the Hardy–Ramanujan number:

The starship Nimbus displays the hull registry number BP-1729.

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"

In one episode, the robot Bender receives a Christmas card from the machine that built him labeled "Son #1729".



The Hardy-Ramanujan number is the smallest product of three distinct primes of the form 6n + 1. [Pol]

The largest number which is divisible by its prime sum of digits (19) and reversal (91)

Ramanujan's famous taxi-cab number (1729 = 123 + 13 = 103 + 93). It is the smallest number expressible as the sum of two positive cubes in two different ways.

The smallest number that is a pseudoprime simultaneously to bases 2, 3 and 5. [Pomerance , Selfridge and Wagstaff]

If you reverse the middle digits of this pseudoprime you get 1279 and 21279 - 1 is a Mersenne prime. [Luhn]

Schiemann's first pair of isospectral lattices L+(1,7,13,19) and L-(1,7,13,19) are of determinant 1*7*13*19 = 1729. [Poo Sung]

The Hardy-Ramanujan number is equal to the average of the only known prime squares of the form n! + 1, i.e., 25, 121, and 5041. [Gudipati]



Interestingly, there are also amicable numbers, evil numbers, happy numbers, hungry numbers, narcisstic numbers, odious numbers, sociable numbers and vampire numbers, just to name a few.


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