taxicab numbers

The number $1729$ has a reputation of its own. The reason is the famous exchange between http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Hardy.htmlG. H. Hardy, a famous British mathematician (1877-1947), and http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Ramanujan.htmlSrinivasa Ramanujan , one of India’s greatest mathematical geniuses (1887-1920):

In 1917, during one visit to Ramanujan in a hospital (he was ill for much of his last three years), Hardy mentioned that the number of the taxi cab that had brought him was $1729$, which, as numbers go, Hardy thought was “rather a dull number”. At this, Ramanujan perked up, and said “No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways.”

Indeed:

$$1729=1+{12}^3=9^3+{10}^3.$$

Moreover, there are other reasons why $1729$ is far from dull. $1729$ is the third Carmichael number. Even more strange, beginning at the $1729$th decimal digit of the transcental number $e$, the next ten successive digits of $e$ are 0719425863. This is the first appearance of all ten digits in a row without repititions.

More generally, the smallest natural number which can be expressed as the sum of $n$ positive cubes is called the $n$th taxicab number. The first taxicab numbers are:

$$2=1^3+1^3,\mathrm{\hspace{0.25em}1729}=1^3+{12}^3=9^3+{10}^3,\mathrm{\hspace{0.25em}87539319}={167}^3+{436}^3={228}^3+{423}^3={255}^3+{414}^3$$

followed by $6963472309248$ (found by E. Rosenstiel, J.A. Dardis, and C.R. Rosenstiel in 1991) and $48988659276962496$ (found by David Wilson on November 21st, 1997).

Title	taxicab numbers
Canonical name	TaxicabNumbers
Date of creation	2013-03-22 15:43:00
Last modified on	2013-03-22 15:43:00
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	6
Author	alozano (2414)
Entry type	Feature
Classification	msc 00A08