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New Upper Bounds for Taxicab and Cabtaxi Numbers
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Christian Boyer

53, rue de Mora

FR-95880 Enghien-les-Bains

France

**Abstract:**

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
different ways".
In memory of this story, this number is now called Taxicab(2) = 1729 =
9^{3} + 10^{3} = 1^{3} + 12^{3},
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:
Cabtaxi(*n*)
is the smallest number
expressible in *n* ways as a sum or difference of two cubes.
For example, Cabtaxi(2) = 91 = 3^{3} + 4^{3} =
6^{3} - 5^{3}.
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).

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(Concerned with sequences
A011541 and
A047696
.)

Received June 26 2007;
revised version received February 22 2008.
Published in *Journal of Integer Sequences*, March 7 2008.

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