JUCS - Journal of Universal Computer Science 9(10): 1196-1203, doi: 10.3217/jucs-009-10-1196

What is the Value of Taxicab(6)?

‡ University of Auckland, Auckland, New Zealand§ Institute of Information Sciences, Massey University at Albany, New Zealand| Department of Computer Science, University of Auckland, Auckland, New Zealand

Corresponding author: Cristian S. Calude ( cristian@cs.auckland.ac.nz ) © Cristian S. Calude, Elena Calude, Michael Dinneen. This article is freely available under the J.UCS Open Content License. Citation:
Calude CSS, Calude E, Dinneen MJ (2003) What is the Value of Taxicab(6)? JUCS - Journal of Universal Computer Science 9(10): 1196-1203. https://doi.org/10.3217/jucs-009-10-1196 |

Abstract

For almost 350 years it was known that 1729 is the smallest integer which can be expressed as the sum of two positive cubes in two different ways. Motivated by a famous story involving Hardy and Ramanujan, a class of numbers called Taxicab Numbers has been defined: Taxicab(k, j, n) is the smallest number which can be expressed as the sum of j kth powers in n different ways. So, Taxicab(3, 2, 2) = 1729, Taxicab(4, 2, 2) = 635318657. Computing Taxicab Numbers is challenging and interesting, both from mathematical and programming points of view. The exact value of Taxicab(6) = Taxicab(3, 2, 6) is not known, however, recent results announced by Rathbun [R2002] show that Taxicab(6) is in the interval [10 18 , 24153319581254312065344]. In this note we show that with probability greater than 99%, Taxicab(6) = 24153319581254312065344.

Keywords

Hardy-Ramanujan Number, Taxicab Number, sampling