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:

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

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 TaxicabNumbers 2013-03-22 15:43:00 2013-03-22 15:43:00 alozano (2414) alozano (2414) 6 alozano (2414) Feature msc 00A08