Taxicab Number

The th taxicab number is the smallest number representable in ways as a sum of positive cubes. The numbers derive their name from the Hardy-Ramanujan number

			(1)
			(2)
			(3)

which is associated with a story told about Ramanujan by G. H. Hardy (Hofstadter 1989, Kanigel 1991, Snow 1993).

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).

However, this property was also known as early as 1657 by F. de Bessy (Berndt and Bhargava 1993, Guy 1994). Leech (1957) found

			(4)
			(5)
			(6)
			(7)

Rosenstiel et al. (1991) recently found

			(8)
			(9)
			(10)
			(11)
			(12)

Wilson (1999) found

			(13)
			(14)
			(15)
			(16)
			(17)
			(18)

The first few taxicab numbers are therefore 2, 1729, 87539319, 6963472309248, 48988659276962496, ... (OEIS A011541).

The sixth taxicab number is

			(19)
			(20)
			(21)
			(22)
			(23)
			(24)
			(25)

(Calude et al. 2003, Hollerbach 2008).

Hardy and Wright (Theorem 412, 1979) show that the number of such sums can be made arbitrarily large but, updating Guy (1994) with Wilson's result, the least example is not known for six or more equal sums.

Sloane defines a slightly different type of taxicab numbers, namely numbers which are sums of two cubes in two or more ways, the first few of which are 1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, ... (OEIS A001235).