New Upper Bounds for Taxicab and Cabtaxi Numbers
53, rue de Mora
Hardy was surprised by Ramanujan's remark about a London
taxi numbered 1729: "it is a very interesting number, it is
the smallest number expressible as a sum of two cubes in two
In memory of this story, this number is now called Taxicab(2) = 1729 =
93 + 103 = 13 + 123,
Taxicab(n) being the smallest number expressible
in n ways as a sum of two cubes.
We can generalize the problem by also allowing differences of cubes:
is the smallest number
expressible in n ways as a sum or difference of two cubes.
For example, Cabtaxi(2) = 91 = 33 + 43 =
63 - 53.
Results were only known up to Taxicab(6) and Cabtaxi(9).
This paper presents a history of the two problems
since Fermat, Frenicle and Viète, and gives new upper bounds for
Taxicab(7) to Taxicab(19), and for
Cabtaxi(10) to Cabtaxi(30).
Decompositions are explicitly given up to Taxicab(12) and Cabtaxi(20).
Full version: pdf,
(Concerned with sequences
Received June 26 2007;
revised version received February 22 2008.
Published in Journal of Integer Sequences, March 7 2008.
Journal of Integer Sequences home page