1729 is sometimes called the Hardy-Ramanujan number. It is the smallest taxicab number, i.e., the smallest number which can be expressed as the sum of two cubes in two different ways:

A more obscure appearance of 1729 is as the average of the greatest member in each pair of (known) Brown numbers (5, 4), (11, 5), and (71, 7):

(K. MacMillan, pers. comm., Apr. 29, 2007).

This property of 1729 was mentioned by the character Robert the sometimes insane mathematician, played by Anthony Hopkins, in the 2005 film Proof. The number 1729 also appeared with no mention of its special property as the number associated with gambler Nick Fisher (Sam Jaeger) in the betting books of The Boss (Morgan Freeman) in the 2006 film Lucky Number Slevin.

1729 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).

REFERENCES:

Georgoulias, T.; Greenwald, S. J.; and Wichterich, M. "Futurama : Mathematics in the Year 3000." Math Horizons, 12-15, Apr. 2004.

Greenwald, S. "Dr. Sarah's Futurama --Mathematics in the Year 3000." https://www.mathsci.appstate.edu/~sjg/futurama/.