# 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 |