CabtaxiNumber 2008-03-21 19:59:29 2008-03-21 19:59:29 Definition taxicab numbers % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A {\em cabtaxi number} for a given $n$ is the smallest positive number which can be written as $a^3 + b^3$ in $n$ different ways, with either $a$ or $b$ allowed to be negative integers. For example, 91 is the 2nd cabtaxi number since it can be expressed $(-5)^3 + 6^3 = 3^3 + 4^3 = 91$. The known cabtaxi numbers are 1, 91, 728, 2741256, 6017193, 1412774811, 11302198488, 137513849003496, 424910390480793000, listed in A047696 of Sloane's OEIS. Adding the restriction $a \geq b > 0$ gives the definition for the taxicab numbers.