Why 1729 (a magic number) is known as Ramanujan Number?

The smallest nontrivial taxicab number, i.e., the smallest number representable in two ways as a sum of two cubes. It is given by:

1729=1³+12³=9³+10³

The story begins like this-

The number derives its name from the following story G. H. Hardy told about Ramanujan. “Once, in the taxi from London, Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, ‘rather a dull number,’ adding that he hoped that wasn’t a bad omen. ‘No, Hardy,’ said Ramanujan, ‘it is a very interesting number. It is the smallest number expressible as the sum of two [positives’ ] cubes in two different ways’ “

Another number which is expressible as the sum of two [positives’ ] cubes in two different ways is-
4104= 16³ + 2³= 15³ + 9³.

This property of 1729 was mentioned by the character Robert the sometimes insane mathematician, played by Anthony Hopkins, in the 2005 film Proof. It was also part of the designation of the spaceship Nimbus BP-1729 appearing in Season 2 of the animated television series Futurama episode DVD 2ACV02 (Greenwald; left figure), as well as the robot character Bender’s serial number, as portrayed in a Christmas card in the episode Xmas Story (Volume 2 DVD, Georgoulias et al. 2004; right figure).

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.

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 and duodecimal.

In base 12, 1729 is written as 1001, so its reciprocal has only period 6 in that base.

1729 has another mildly interesting property: the 1729th decimal place is the beginning of the first consecutive occurrence of all ten digits without repetition 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:

1 + 7 + 2 + 9 = 19
19 × 91 = 1729

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

Reference: Wikipidea

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